[{"data":1,"prerenderedAt":1055},["ShallowReactive",2],{"questions:gravity":3},{"bank":4,"contentHash":1043,"dependencyHashes":1044,"releaseId":1054},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1030,"reviewStatus":1039,"authoring":1040},1,"gravity","Gravity question bank","Seventy questions spanning mass and weight, free fall, air resistance, Newton's law, orbits and escape velocity, ISRO's missions, weightlessness, pendulums and the edge of general relativity. Every numeric answer was computed in Python and checked before being written here — work through the levels in order, or dip into whichever section you are studying.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"mass-weight","Mass and weight","Telling mass and weight apart, and using weight = mass × g on other worlds.",{"id":15,"title":16,"description":17},"free-fall","Free fall","Distance and speed during a fall with no air resistance: d = ½ g t² and v = g t.",{"id":19,"title":20,"description":21},"air-resistance","Air resistance and terminal velocity","How drag changes a fall, and why shape and mass both matter in air.",{"id":23,"title":24,"description":25},"newtons-law","Newton’s law of gravitation","F = G m1 m2 ÷ r² and the inverse-square fall-off with distance.",{"id":27,"title":28,"description":29},"orbits","Orbits, escape and Kepler","Orbital speed, escape velocity, and how period relates to distance.",{"id":31,"title":32,"description":33},"satellites-isro","Satellites and ISRO missions","The ISS, geostationary orbit, and the arithmetic behind Chandrayaan-3 and Mangalyaan.",{"id":35,"title":36,"description":37},"weightlessness","Weightlessness","Why astronauts float, and telling real free fall from ordinary flight.",{"id":39,"title":40,"description":41},"pendulum","The pendulum","T = 2π√(L\u002Fg) and using a pendulum to measure g.",{"id":43,"title":44,"description":45},"history-extend","History and the bigger picture","Galileo, Einstein, general relativity and black holes.",[47,63,88,107,118,126,146,160,171,183,195,216,225,235,245,256,266,277,289,309,329,341,350,369,380,398,407,426,445,458,476,496,507,518,527,537,556,575,586,598,617,630,641,653,665,676,695,713,725,734,754,765,784,794,813,828,840,858,867,877,896,915,924,934,946,955,966,986,998,1017],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":56,"solution":59,"skills":60},"gravity.q001","foundation","A 10 kg bag of rice is taken to the Moon, where g = 1.62 N\u002Fkg. What does it weigh there, in newtons?",{"kind":52,"answer":53,"tolerance":54,"unit":55},"number",16.2,0.1,"N",[57,58],"Use weight = mass × g.","10 × 1.62.","weight = 10 × 1.62 = **16.2 N**. The mass is still 10 kg — only the weight changes.",[61,62],"weight=mass×g","other worlds",{"id":64,"section":11,"level":49,"prompt":65,"check":66,"hints":82,"solution":84,"skills":85},"gravity.q002","Which of these is the correct value of g on Mars?",{"kind":67,"options":68,"correct":81},"choice",[69,72,75,78],{"id":70,"label":71},"a","9.8 N\u002Fkg",{"id":73,"label":74},"b","3.72 N\u002Fkg",{"id":76,"label":77},"c","1.62 N\u002Fkg",{"id":79,"label":80},"d","24.79 N\u002Fkg",[73],[83],"Mars is smaller and less massive than Earth, so its pull is weaker than Earth’s but stronger than the Moon’s.","Mars’s surface gravity is **3.72 N\u002Fkg**, a little over a third of Earth’s 9.8 N\u002Fkg. (a) is Earth, (c) is the Moon, (d) is Jupiter.",[86,87],"g values","planets",{"id":89,"section":11,"level":49,"prompt":90,"check":91,"hints":102,"solution":104,"skills":105},"gravity.q003","A 40 kg child travels from Earth to Jupiter’s cloud tops. What is the child’s mass there?",{"kind":67,"options":92,"correct":101},[93,95,97,99],{"id":70,"label":94},"40 kg",{"id":73,"label":96},"991.6 kg",{"id":76,"label":98},"24.79 kg",{"id":79,"label":100},"162 kg",[70],[103],"Mass is the amount of matter in the child, not the pull on it.","**40 kg.** Mass never changes with travel. 991.6 N would be the child’s *weight* on Jupiter (40 × 24.79).",[106],"mass vs weight",{"id":108,"section":11,"level":109,"prompt":110,"check":111,"hints":114,"solution":116,"skills":117},"gravity.q004","core","A 25 kg sample container is taken to Mars, where g = 3.72 N\u002Fkg. What does it weigh there, in newtons?",{"kind":52,"answer":112,"tolerance":113,"unit":55},93,0.5,[115],"weight = mass × g.","weight = 25 × 3.72 = **93.0 N**.",[61],{"id":119,"section":11,"level":109,"prompt":120,"check":121,"hints":123,"solution":124,"skills":125},"gravity.q005","An astronaut with all her equipment has a mass of 65 kg. What is her weight on Earth, in newtons? (g = 9.8 N\u002Fkg)",{"kind":52,"answer":122,"tolerance":5,"unit":55},637,[115],"weight = 65 × 9.8 = **637.0 N**.",[61],{"id":127,"section":11,"level":109,"prompt":128,"check":129,"hints":140,"solution":142,"skills":143},"gravity.q006","Which instrument would give the same reading for an object’s mass on Earth and on the Moon?",{"kind":67,"options":130,"correct":139},[131,133,135,137],{"id":70,"label":132},"A spring bathroom scale",{"id":73,"label":134},"A pan balance comparing it against known masses",{"id":76,"label":136},"A newton-meter",{"id":79,"label":138},"None of these — mass cannot be measured",[73],[141],"A spring scale measures force (weight); a pan balance compares masses directly, and gravity affects both pans equally.","**A pan balance.** Gravity pulls on both pans equally, whatever g is, so it always compares masses fairly. A spring scale measures force, so it reads differently under different gravity.",[144,145],"measuring mass","balances",{"id":147,"section":11,"level":148,"prompt":149,"check":150,"hints":154,"solution":157,"skills":158},"gravity.q007","stretch","A 65 kg astronaut steps onto an Earth-calibrated spring bathroom scale while visiting a lunar base (g = 1.62 N\u002Fkg). What number, in kilograms, would the scale display? Give your answer to two decimal places.",{"kind":52,"answer":151,"tolerance":152,"unit":153},10.74,0.05,"kg",[155,156],"The scale reads a force and divides by Earth’s g = 9.8 to print \"kilograms\".","Find the real Moon weight first: 65 × 1.62.","Real weight on the Moon: 65 × 1.62 = 105.3 N. The scale divides by 9.8: 105.3 ÷ 9.8 ≈ **10.74 kg**. The astronaut’s real mass is still 65 kg — the scale has simply been fooled.",[61,159],"measurement errors",{"id":161,"section":11,"level":148,"prompt":162,"check":163,"hints":166,"solution":168,"skills":169},"gravity.q008","A suitcase weighs 297.5 N on Jupiter’s cloud tops, where g = 24.79 N\u002Fkg. What is its mass, in kilograms?",{"kind":52,"answer":164,"tolerance":165,"unit":153},12,0.2,[167],"Rearrange weight = mass × g to mass = weight ÷ g.","mass = 297.5 ÷ 24.79 = **12.0 kg**.",[61,170],"rearranging formulas",{"id":172,"section":11,"level":173,"prompt":174,"check":175,"hints":177,"solution":180,"skills":181},"gravity.q009","challenge","A rock weighs 93.0 N on Mars (g = 3.72 N\u002Fkg). What would the same rock weigh on the Moon (g = 1.62 N\u002Fkg)? Work out its mass first.",{"kind":52,"answer":176,"tolerance":113,"unit":55},40.5,[178,179],"First find the mass: mass = 93.0 ÷ 3.72.","Then use that mass with the Moon’s g.","Mass = 93.0 ÷ 3.72 = 25 kg. Weight on the Moon = 25 × 1.62 = **40.5 N**. (The Mars weight is about 2.3 times the Moon weight, matching the ratio of g values: 3.72 ÷ 1.62 ≈ 2.3.)",[61,182],"two-step reasoning",{"id":184,"section":15,"level":49,"prompt":185,"check":186,"hints":189,"solution":191,"skills":192},"gravity.q010","Ignoring air resistance, how far does an object fall from rest in 1 second on Earth? (g = 9.8 m\u002Fs²)",{"kind":52,"answer":187,"tolerance":152,"unit":188},4.9,"m",[190],"Use d = ½ × g × t², with t = 1.","d = ½ × 9.8 × 1² = **4.9 m**.",[193,194],"free fall","d=½gt²",{"id":196,"section":15,"level":49,"prompt":197,"check":198,"hints":209,"solution":212,"skills":213},"gravity.q011","A stone falls freely for 2 seconds. What is its speed at that moment?",{"kind":67,"options":199,"correct":208},[200,202,204,206],{"id":70,"label":201},"9.8 m\u002Fs",{"id":73,"label":203},"19.6 m\u002Fs",{"id":76,"label":205},"19.6 m",{"id":79,"label":207},"4.9 m\u002Fs",[73],[210,211],"Use v = g × t.","Careful with the unit — speed, not distance.","v = 9.8 × 2 = **19.6 m\u002Fs**. (Option c has the right number but the wrong unit — 19.6 m is actually the *distance* fallen at that same moment, a coincidence that only happens at t = 2 s.)",[214,215],"v=gt","units",{"id":217,"section":15,"level":109,"prompt":218,"check":219,"hints":221,"solution":223,"skills":224},"gravity.q012","Ignoring air resistance, how far does an object fall from rest in 4 seconds on Earth?",{"kind":52,"answer":220,"tolerance":54,"unit":188},78.4,[222],"d = ½ × 9.8 × t², with t = 4.","d = ½ × 9.8 × 4² = 4.9 × 16 = **78.4 m**.",[194],{"id":226,"section":15,"level":109,"prompt":227,"check":228,"hints":231,"solution":233,"skills":234},"gravity.q013","Ignoring air resistance, what is the speed of an object after falling for 5 seconds? Give your answer in m\u002Fs.",{"kind":52,"answer":229,"tolerance":54,"unit":230},49,"m\u002Fs",[232],"v = g × t.","v = 9.8 × 5 = **49.0 m\u002Fs** (about 176.4 km\u002Fh).",[214],{"id":236,"section":15,"level":109,"prompt":237,"check":238,"hints":240,"solution":242,"skills":243},"gravity.q014","A stone is dropped and takes 4.04 seconds to hit the ground. Roughly how high, in metres, was it dropped from? (No air resistance; round to the nearest metre.)",{"kind":52,"answer":239,"tolerance":5,"unit":188},80,[241],"Use d = ½ × g × t² with t = 4.04.","d = ½ × 9.8 × 4.04² ≈ **80 m**. (This is the reverse of the fall-time formula: t = √(2d÷g).)",[194,244],"working backwards",{"id":246,"section":15,"level":148,"prompt":247,"check":248,"hints":250,"solution":253,"skills":254},"gravity.q015","A ball is dropped from a height of 80 m. Ignoring air resistance, how fast is it moving when it lands? Give your answer in m\u002Fs.",{"kind":52,"answer":249,"tolerance":165,"unit":230},39.6,[251,252],"Use v = √(2 × g × d).","Or find the time first with t = √(2d÷g), then v = g × t.","v = √(2 × 9.8 × 80) = √1,568 ≈ **39.6 m\u002Fs** (about 142.6 km\u002Fh).",[255],"v=√(2gd)",{"id":257,"section":15,"level":148,"prompt":258,"check":259,"hints":261,"solution":263,"skills":264},"gravity.q016","A freely falling object has already fallen 122.5 m after 5 seconds. How far does it fall during the 6th second alone (that is, between t = 5 s and t = 6 s)?",{"kind":52,"answer":260,"tolerance":165,"unit":188},53.9,[262],"Find the total distance at t = 6 s, then subtract the distance at t = 5 s.","d(6) = 4.9 × 36 = 176.4 m. d(5) = 122.5 m. Difference = 176.4 − 122.5 = **53.9 m**. This continues Galileo’s odd-number pattern (1, 3, 5, 7, 9, 11 units of 4.9 m).",[194,265],"Galileo odd numbers",{"id":267,"section":15,"level":148,"prompt":268,"check":269,"hints":272,"solution":274,"skills":275},"gravity.q017","On Mars, g = 3.72 m\u002Fs². How long, in seconds, does it take an object to fall 50 m from rest, ignoring air resistance?",{"kind":52,"answer":270,"tolerance":152,"unit":271},5.18,"s",[273],"Rearrange d = ½ g t² to t = √(2d ÷ g).","t = √(2 × 50 ÷ 3.72) = √26.88 ≈ **5.18 s**.",[276,62],"t=√(2d\u002Fg)",{"id":278,"section":15,"level":173,"prompt":279,"check":280,"hints":283,"solution":286,"skills":287},"gravity.q018","A stone is dropped from a cliff on Earth. After how many whole seconds will it first have fallen more than 100 m? (Ignore air resistance; find the answer by testing whole-second values.)",{"kind":52,"answer":281,"tolerance":282,"unit":271},5,0,[284,285],"Try t = 4: d = 4.9 × 16 = 78.4 m — not yet 100 m.","Try t = 5.","At t = 4 s, d = 78.4 m (not enough). At **t = 5 s**, d = 4.9 × 25 = 122.5 m, which is the first whole second where the distance passes 100 m.",[194,288],"testing values",{"id":290,"section":19,"level":49,"prompt":291,"check":292,"hints":303,"solution":305,"skills":306},"gravity.q019","What is \"terminal velocity\"?",{"kind":67,"options":293,"correct":302},[294,296,298,300],{"id":70,"label":295},"The fastest speed anything can ever fall at",{"id":73,"label":297},"The steady speed reached when air resistance grows to equal weight",{"id":76,"label":299},"The speed at the very start of a fall",{"id":79,"label":301},"A speed that only applies in space",[73],[304],"Think about the balance of two forces: gravity pulling down, air resistance pushing up.","**(b).** Once drag has grown enough to exactly balance weight, there is no net force left, so the falling object stops speeding up and continues at a constant, \"terminal\", speed.",[307,308],"terminal velocity","definitions",{"id":310,"section":19,"level":49,"prompt":311,"check":312,"hints":323,"solution":325,"skills":326},"gravity.q020","A flat sheet of paper and the same sheet crumpled into a ball are dropped together. What happens, and why?",{"kind":67,"options":313,"correct":322},[314,316,318,320],{"id":70,"label":315},"They land together, because crumpling does not change the mass",{"id":73,"label":317},"The flat sheet wins, because it is lighter",{"id":76,"label":319},"The crumpled ball wins, because it has less air resistance for the same mass",{"id":79,"label":321},"They float, because paper never really falls",[76],[324],"Not one atom was added or removed by crumpling — only the shape changed.","**(c).** Same mass, but the crumpled ball presents far less area to the air, so it experiences much less drag and reaches a higher speed before drag catches up with its (unchanged) weight.",[327,328],"air resistance","fair tests",{"id":330,"section":19,"level":109,"prompt":331,"check":332,"hints":335,"solution":337,"skills":338},"gravity.q021","A skydiver’s terminal velocity, falling belly-down, is about 55 m\u002Fs. Convert this to km\u002Fh.",{"kind":52,"answer":333,"tolerance":5,"unit":334},198,"km\u002Fh",[336],"Multiply by 3.6 to convert m\u002Fs to km\u002Fh.","55 × 3.6 = **198 km\u002Fh**.",[339,340],"unit conversion","m\u002Fs to km\u002Fh",{"id":342,"section":19,"level":109,"prompt":343,"check":344,"hints":346,"solution":348,"skills":349},"gravity.q022","Under an open parachute, a skydiver’s terminal velocity is about 5.5 m\u002Fs. What is this in km\u002Fh?",{"kind":52,"answer":345,"tolerance":165,"unit":334},19.8,[347],"Multiply by 3.6.","5.5 × 3.6 = **19.8 km\u002Fh** — about the speed of jumping off a low wall.",[339],{"id":351,"section":19,"level":109,"prompt":352,"check":353,"hints":364,"solution":366,"skills":367},"gravity.q023","Two skydivers, of different masses but with identical parachutes and body positions, jump together. Who has the higher terminal velocity?",{"kind":67,"options":354,"correct":363},[355,357,359,361],{"id":70,"label":356},"The lighter one",{"id":73,"label":358},"The heavier one",{"id":76,"label":360},"They are exactly the same",{"id":79,"label":362},"It depends only on the weather",[73],[365],"Same shape means the same drag at any given speed — but the two have different weights for that drag to balance.","**The heavier one.** Identical shape gives identical drag at a given speed, but the heavier skydiver needs a higher speed before drag grows enough to balance her greater weight.",[307,368],"reasoning",{"id":370,"section":19,"level":148,"prompt":371,"check":372,"hints":375,"solution":377,"skills":378},"gravity.q024","If a raindrop fell 3 km from a cloud with absolutely no air resistance, roughly how fast (in m\u002Fs) would it be moving when it hit the ground?",{"kind":52,"answer":373,"tolerance":374,"unit":230},242,3,[376],"Use v = √(2 × g × d), with d = 3,000 m.","v = √(2 × 9.8 × 3,000) = √58,800 ≈ **242 m\u002Fs** (about 873 km\u002Fh) — faster than an airliner. Air resistance is what actually keeps real raindrops down to about 6.5 m\u002Fs.",[255,379],"why air resistance matters",{"id":381,"section":19,"level":148,"prompt":382,"check":383,"hints":394,"solution":396,"skills":397},"gravity.q025","Real raindrops fall at only about 6.5 m\u002Fs (23 km\u002Fh), even though a 3 km fall with no air resistance would give about 242 m\u002Fs. What does this tell you?",{"kind":67,"options":384,"correct":393},[385,387,389,391],{"id":70,"label":386},"Gravity is much weaker near clouds",{"id":73,"label":388},"Raindrops reach terminal velocity very early in the fall, long before the ground",{"id":76,"label":390},"Raindrops are not really affected by gravity at all",{"id":79,"label":392},"The 242 m\u002Fs calculation must be wrong",[73],[395],"A small, light drop has very little weight for drag to balance, so the balance point is reached quickly, at low speed.","**(b).** A small, light raindrop needs only a small amount of drag to balance its weight, so it reaches terminal velocity within the first few metres and then falls at a steady, safe speed the rest of the way, whatever the total height of the cloud.",[307,368],{"id":399,"section":19,"level":173,"prompt":400,"check":401,"hints":402,"solution":404,"skills":405},"gravity.q026","A hailstone’s real terminal velocity is about 20 m\u002Fs. If it fell 2 km from a cloud with no air resistance at all, roughly how fast (in m\u002Fs) would it be moving instead? By what rough factor would that be faster than its real terminal velocity? Give the no-drag speed only, to the nearest whole number.",{"kind":52,"answer":333,"tolerance":374,"unit":230},[403],"Use v = √(2 × g × d) with d = 2,000 m, ignoring the real terminal velocity for this part.","v = √(2 × 9.8 × 2,000) = √39,200 ≈ **198 m\u002Fs**. That is about 10 times the hailstone’s real terminal velocity of 20 m\u002Fs — showing how much work air resistance is doing.",[255,406],"comparing scenarios",{"id":408,"section":23,"level":49,"prompt":409,"check":410,"hints":421,"solution":423,"skills":424},"gravity.q027","According to Newton’s law of gravitation, the force between two objects gets stronger when...",{"kind":67,"options":411,"correct":420},[412,414,416,418],{"id":70,"label":413},"their masses are bigger and they are closer together",{"id":73,"label":415},"their masses are bigger and they are further apart",{"id":76,"label":417},"their masses are smaller and they are closer together",{"id":79,"label":419},"distance is the only thing that matters",[70],[422],"Two separate rules combine: more mass means more pull; more distance means less pull.","**(a).** Force grows with bigger masses and shrinks with distance — so it is strongest for big masses that are close together.",[425,308],"Newton’s law",{"id":427,"section":23,"level":49,"prompt":428,"check":429,"hints":440,"solution":442,"skills":443},"gravity.q028","If the distance between two masses is doubled, what happens to the gravitational force between them?",{"kind":67,"options":430,"correct":439},[431,433,435,437],{"id":70,"label":432},"It doubles",{"id":73,"label":434},"It halves",{"id":76,"label":436},"It becomes one quarter",{"id":79,"label":438},"It stays the same",[76],[441],"The force depends on 1 ÷ distance², not 1 ÷ distance.","Doubling the distance means dividing the force by 2² = 4, so it becomes **one quarter**.",[444],"inverse-square law",{"id":446,"section":23,"level":109,"prompt":447,"check":448,"hints":452,"solution":455,"skills":456},"gravity.q029","Two objects of mass 100 kg each are placed 1 metre apart. Using F = G × m1 × m2 ÷ r² with G = 6.674 × 10⁻¹¹ N·m²\u002Fkg², find the force between them. Give your answer in units of 10⁻⁷ N (i.e. just the number in front).",{"kind":52,"answer":449,"tolerance":450,"unit":451},6.674,0.01,"×10⁻⁷ N",[453,454],"F = 6.674 × 10⁻¹¹ × 100 × 100 ÷ 1².","100 × 100 = 10,000 = 10⁴.","F = 6.674 × 10⁻¹¹ × 10⁴ ÷ 1 = 6.674 × 10⁻⁷ N, so the answer is **6.674** (× 10⁻⁷ N).",[425,457],"F=Gm1m2\u002Fr²",{"id":459,"section":23,"level":109,"prompt":460,"check":461,"hints":472,"solution":474,"skills":475},"gravity.q030","At three times the original distance, the gravitational force between two masses becomes...",{"kind":67,"options":462,"correct":471},[463,465,467,469],{"id":70,"label":464},"one third",{"id":73,"label":466},"one sixth",{"id":76,"label":468},"one ninth",{"id":79,"label":470},"unchanged",[76],[473],"1 ÷ 3² = 1 ÷ 9.","Force ∝ 1 ÷ distance², so tripling the distance gives 1 ÷ 3² = **one ninth** of the original force.",[444],{"id":477,"section":23,"level":109,"prompt":478,"check":479,"hints":490,"solution":492,"skills":493},"gravity.q031","A 40 kg child pulls on the Earth with some gravitational force. How does the Earth’s pull on the child compare?",{"kind":67,"options":480,"correct":489},[481,483,485,487],{"id":70,"label":482},"The Earth pulls on the child with exactly the same size of force",{"id":73,"label":484},"The Earth pulls with a much smaller force, since the child is so small",{"id":76,"label":486},"The Earth does not pull on the child at all; only the child pulls on the Earth",{"id":79,"label":488},"The forces are unrelated",[70],[491],"Gravitational forces always come in equal, opposite pairs.","**(a).** Every gravitational pull comes as an equal, opposite pair: the child pulls the Earth up exactly as hard as the Earth pulls the child down. The huge difference in *effect* comes from the huge difference in mass, not in force.",[494,495],"Newton’s third law","equal and opposite",{"id":497,"section":23,"level":148,"prompt":498,"check":499,"hints":502,"solution":505,"skills":506},"gravity.q032","Two masses of 300 kg and 400 kg are placed 2 metres apart. Using F = G × m1 × m2 ÷ r², find the force between them, in units of 10⁻⁶ N.",{"kind":52,"answer":500,"tolerance":450,"unit":501},2.0022,"×10⁻⁶ N",[503,504],"F = 6.674 × 10⁻¹¹ × 300 × 400 ÷ 2².","300 × 400 = 120,000; 2² = 4.","F = 6.674 × 10⁻¹¹ × 120,000 ÷ 4 = 6.674 × 10⁻¹¹ × 30,000 = 2.0022 × 10⁻⁶ N, so the answer is **2.0022** (× 10⁻⁶ N).",[425,457],{"id":508,"section":23,"level":148,"prompt":509,"check":510,"hints":513,"solution":515,"skills":516},"gravity.q033","At 3 Earth radii from the centre, what is g, in N\u002Fkg? (g at the surface, 1 radius out, is 9.8 N\u002Fkg.)",{"kind":52,"answer":511,"tolerance":450,"unit":512},1.089,"N\u002Fkg",[514],"Use the inverse-square law: g at n radii = 9.8 ÷ n².","g = 9.8 ÷ 3² = 9.8 ÷ 9 ≈ **1.089 N\u002Fkg**.",[444,517],"g vs distance",{"id":519,"section":23,"level":148,"prompt":520,"check":521,"hints":523,"solution":525,"skills":526},"gravity.q034","At 5 Earth radii from the centre, what is g, in N\u002Fkg?",{"kind":52,"answer":522,"tolerance":450,"unit":512},0.392,[524],"g at n radii = 9.8 ÷ n².","g = 9.8 ÷ 5² = 9.8 ÷ 25 = **0.392 N\u002Fkg**.",[444],{"id":528,"section":23,"level":173,"prompt":529,"check":530,"hints":532,"solution":535,"skills":536},"gravity.q035","Two masses of 1,000 kg and 2,000 kg attract each other with a force of about 1.335 × 10⁻⁶ N. Using F = G × m1 × m2 ÷ r², work out the distance between them, in metres.",{"kind":52,"answer":531,"tolerance":165,"unit":188},10,[533,534],"Rearrange to r² = G × m1 × m2 ÷ F, then take the square root.","G × m1 × m2 = 6.674 × 10⁻¹¹ × 1,000 × 2,000.","r² = (6.674 × 10⁻¹¹ × 1,000 × 2,000) ÷ (1.335 × 10⁻⁶) = (1.3348 × 10⁻⁴) ÷ (1.335 × 10⁻⁶) ≈ 100, so r = √100 = **10 m**.",[425,170],{"id":538,"section":27,"level":49,"prompt":539,"check":540,"hints":551,"solution":553,"skills":554},"gravity.q036","Roughly what speed does an object need to orbit just above Earth’s surface (ignoring air)?",{"kind":67,"options":541,"correct":550},[542,544,546,548],{"id":70,"label":543},"About 7.9 km\u002Fs",{"id":73,"label":545},"About 0.79 km\u002Fs",{"id":76,"label":547},"About 79 km\u002Fs",{"id":79,"label":549},"About 340 m\u002Fs",[70],[552],"This is Newton’s cannon speed: fast enough that the ground curves away as quickly as you fall.","About **7.9 km\u002Fs** (roughly 28,400 km\u002Fh) — the speed at which falling and the Earth’s curve away happen at exactly the same rate.",[555],"orbital speed",{"id":557,"section":27,"level":49,"prompt":558,"check":559,"hints":570,"solution":572,"skills":573},"gravity.q037","Escape velocity is always related to circular orbital speed by which factor?",{"kind":67,"options":560,"correct":569},[561,563,565,567],{"id":70,"label":562},"√2 times bigger",{"id":73,"label":564},"Exactly the same",{"id":76,"label":566},"Half as big",{"id":79,"label":568},"10 times bigger",[70],[571],"This comes from comparing the force and energy arguments for orbiting and escaping.","Escape velocity = **√2 × orbital speed**, always, for any world — about 11.2 km\u002Fs from Earth, compared with 7.9 km\u002Fs to orbit.",[574],"escape velocity",{"id":576,"section":27,"level":109,"prompt":577,"check":578,"hints":581,"solution":583,"skills":584},"gravity.q038","A geostationary satellite orbits at 35,786 km above the equator. Roughly how many hours does it take to complete one orbit?",{"kind":52,"answer":579,"tolerance":54,"unit":580},23.93,"h",[582],"A geostationary orbit is designed to match the time the Earth takes to spin once.","About **23.93 hours** (23 h 56 min) — exactly the time the Earth takes to rotate once relative to the stars, which is why the satellite appears to stay in the same spot in the sky.",[585],"geostationary orbit",{"id":587,"section":27,"level":109,"prompt":588,"check":589,"hints":592,"solution":594,"skills":595},"gravity.q039","The ISS takes about 92.4 minutes to complete one orbit. Roughly how many complete orbits does it make in 24 hours?",{"kind":52,"answer":590,"tolerance":591,"unit":27},15.6,0.3,[593],"24 hours = 1,440 minutes. Divide by 92.4.","1,440 ÷ 92.4 ≈ **15.6 orbits** a day — which is why NASA describes the crew as seeing about 16 sunrises daily.",[596,597],"orbital period","division",{"id":599,"section":27,"level":109,"prompt":600,"check":601,"hints":612,"solution":614,"skills":615},"gravity.q040","Which of these three orbits its central body the fastest: the ISS (400 km up), a geostationary satellite (35,786 km up), or the Moon (384,400 km away)?",{"kind":67,"options":602,"correct":611},[603,605,607,609],{"id":70,"label":604},"The ISS",{"id":73,"label":606},"The geostationary satellite",{"id":76,"label":608},"The Moon",{"id":79,"label":610},"They all move at the same speed",[70],[613],"Closer orbits need higher speeds to keep \"missing\" the planet.","**The ISS**, at about 7.7 km\u002Fs. A geostationary satellite moves at about 3.07 km\u002Fs, and the Moon at only about 1 km\u002Fs — higher orbits are always slower, not faster.",[616],"orbital speed vs altitude",{"id":618,"section":27,"level":148,"prompt":619,"check":620,"hints":624,"solution":627,"skills":628},"gravity.q041","Using v = √(GM ÷ r) with GM = 3.986 × 10¹⁴ m³\u002Fs², find the circular orbital speed at r = 7,000 km (in km\u002Fs, to two decimal places).",{"kind":52,"answer":621,"tolerance":622,"unit":623},7.55,0.03,"km\u002Fs",[625,626],"Convert r to metres first: 7,000 km = 7,000,000 m.","v = √(3.986 × 10¹⁴ ÷ 7,000,000).","v = √(3.986 × 10¹⁴ ÷ 7 × 10⁶) = √(5.694 × 10⁷) ≈ **7.55 km\u002Fs**.",[629],"v=√(GM\u002Fr)",{"id":631,"section":27,"level":148,"prompt":632,"check":633,"hints":635,"solution":638,"skills":639},"gravity.q042","Using v = √(2GM ÷ r) with GM = 3.986 × 10¹⁴ m³\u002Fs², find the escape velocity at r = 8,000 km (in km\u002Fs, to two decimal places).",{"kind":52,"answer":634,"tolerance":622,"unit":623},9.98,[636,637],"Convert r to metres: 8,000 km = 8,000,000 m.","v = √(2 × 3.986 × 10¹⁴ ÷ 8,000,000).","v = √(2 × 3.986 × 10¹⁴ ÷ 8 × 10⁶) = √(9.965 × 10⁷) ≈ **9.98 km\u002Fs**.",[640],"v=√(2GM\u002Fr)",{"id":642,"section":27,"level":148,"prompt":643,"check":644,"hints":647,"solution":650,"skills":651},"gravity.q043","The orbital period at r = 7,000 km is about how many minutes? (Use T = 2πr ÷ v, with v = 7.55 km\u002Fs from an earlier question.)",{"kind":52,"answer":645,"tolerance":5,"unit":646},97.1,"min",[648,649],"T = 2πr ÷ v. Convert everything to consistent units (metres and m\u002Fs, or km and km\u002Fs).","2 × π × 7,000 ÷ 7.55 gives the period in seconds; divide by 60 for minutes.","T = (2 × π × 7,000) ÷ 7.55 ≈ 5,827 s ≈ **97.1 minutes**.",[652],"T=2πr\u002Fv",{"id":654,"section":27,"level":173,"prompt":655,"check":656,"hints":658,"solution":661,"skills":662},"gravity.q044","A satellite orbits at the ISS’s radius (6,771 km) with a period of 92.4 minutes. Using Kepler’s third law (T ∝ r^1.5), estimate the period, in minutes, of a satellite at double that radius (13,542 km).",{"kind":52,"answer":657,"tolerance":374,"unit":646},261.4,[659,660],"If radius doubles, period is multiplied by 2^1.5 (2 × √2).","2 × √2 ≈ 2.828.","New period = 92.4 × 2^1.5 = 92.4 × 2.828 ≈ **261.4 minutes** (about 4 h 21 min).",[663,664],"Kepler’s third law","scaling",{"id":666,"section":27,"level":173,"prompt":667,"check":668,"hints":671,"solution":673,"skills":674},"gravity.q045","A small moon’s circular orbital speed just above its parent planet’s surface is 3.5 km\u002Fs. Using escape velocity = √2 × orbital speed, what is the escape velocity from that surface, in km\u002Fs, to two decimal places?",{"kind":52,"answer":669,"tolerance":670,"unit":623},4.95,0.02,[672],"Multiply 3.5 by √2 ≈ 1.4142.","3.5 × 1.4142 ≈ **4.95 km\u002Fs**.",[574,675],"ratios",{"id":677,"section":31,"level":49,"prompt":678,"check":679,"hints":690,"solution":692,"skills":693},"gravity.q046","Roughly how high above the Earth does the International Space Station orbit?",{"kind":67,"options":680,"correct":689},[681,683,685,687],{"id":70,"label":682},"About 400 km",{"id":73,"label":684},"About 4,000 km",{"id":76,"label":686},"About 40 km",{"id":79,"label":688},"About 35,786 km",[70],[691],"This is \"low Earth orbit\" — high enough to avoid most of the atmosphere, low enough for a fast orbit.","About **400 km** — low Earth orbit. (d) is the very different, much higher geostationary altitude.",[694],"ISS altitude",{"id":696,"section":31,"level":49,"prompt":697,"check":698,"hints":709,"solution":711,"skills":712},"gravity.q047","Why does a satellite in geostationary orbit appear to stay fixed in the sky?",{"kind":67,"options":699,"correct":708},[700,702,704,706],{"id":70,"label":701},"It is not really moving at all",{"id":73,"label":703},"Its orbital period exactly matches the time the Earth takes to spin once",{"id":76,"label":705},"It is attached to the ground by a cable",{"id":79,"label":707},"It uses its engines constantly to hover",[73],[710],"Think about what \"keeping pace with the ground\" would require.","**(b).** At 35,786 km above the equator, one orbit takes 23 h 56 min — exactly Earth’s rotation period — so the satellite and the ground below stay in step.",[585],{"id":714,"section":31,"level":109,"prompt":715,"check":716,"hints":719,"solution":721,"skills":722},"gravity.q048","Chandrayaan-3 launched on 14 July 2023 and its Vikram lander touched down near the lunar south pole on 23 August 2023. How many days after launch did it land?",{"kind":52,"answer":717,"tolerance":282,"unit":718},40,"days",[720],"Count the days from 14 July to 23 August.","From 14 July to 23 August 2023 is **40 days**.",[723,724],"ISRO missions","date arithmetic",{"id":726,"section":31,"level":109,"prompt":727,"check":728,"hints":730,"solution":732,"skills":733},"gravity.q049","Mangalyaan (the Mars Orbiter Mission) launched on 5 November 2013 and entered orbit around Mars on 24 September 2014. How many days after launch did it arrive?",{"kind":52,"answer":729,"tolerance":282,"unit":718},323,[731],"Count from 5 November 2013 to 24 September 2014 — nearly a year.","From 5 Nov 2013 to 24 Sep 2014 is **323 days** — about a month of Earth-orbit-raising plus roughly ten months coasting to Mars.",[723,724],{"id":735,"section":31,"level":109,"prompt":736,"check":737,"hints":748,"solution":750,"skills":751},"gravity.q050","Why did Mangalyaan use six separate Earth-orbit-raising burns instead of one large burn straight to Mars?",{"kind":67,"options":738,"correct":747},[739,741,743,745],{"id":70,"label":740},"It was an accident caused by an engine fault",{"id":73,"label":742},"A smaller, cheaper rocket could launch it, with its own engine gradually building up speed instead",{"id":76,"label":744},"Mars was not in the right position for a direct route",{"id":79,"label":746},"It had no way to measure its speed accurately",[73],[749],"Think about what a single, huge burn straight from launch would require of the rocket.","**(b).** A single huge burn needs a much larger, more expensive rocket. Multiple smaller burns, spread across several orbits, let a modest rocket do the launch while the spacecraft’s own engine gradually raised its orbit — a major reason the mission was so cost-effective.",[752,753],"ISRO strategy","orbit-raising",{"id":755,"section":31,"level":148,"prompt":756,"check":757,"hints":761,"solution":763,"skills":764},"gravity.q051","Chandrayaan-3’s lander arrived at the Moon travelling at roughly 1,680 m\u002Fs and had to slow to under 2 m\u002Fs to land safely. Roughly by what factor did its speed need to be reduced?",{"kind":52,"answer":758,"tolerance":759,"unit":760},840,20,"×",[762],"Divide the starting speed by the target speed: 1,680 ÷ 2.","1,680 ÷ 2 = **840 times** slower — all achieved by the lander’s own engines, with no atmosphere to help and no possibility of a second attempt.",[723,675],{"id":766,"section":31,"level":148,"prompt":767,"check":768,"hints":779,"solution":781,"skills":782},"gravity.q052","A satellite is moved from a 400 km orbit to a much higher 20,000 km orbit. What happens to its orbital speed?",{"kind":67,"options":769,"correct":778},[770,772,774,776],{"id":70,"label":771},"It increases, because higher orbits need more energy",{"id":73,"label":773},"It decreases, because gravity is weaker further out",{"id":76,"label":775},"It stays exactly the same",{"id":79,"label":777},"It becomes zero",[73],[780],"Compare the ISS (400 km, 7.7 km\u002Fs) with a geostationary satellite (35,786 km, 3.07 km\u002Fs).","**(b).** Further from Earth, gravity is weaker, so less sideways speed is needed to stay in a circular orbit — even though *getting* to the higher orbit requires extra energy from the rocket.",[616,783],"common mistake",{"id":785,"section":31,"level":173,"prompt":786,"check":787,"hints":789,"solution":791,"skills":792},"gravity.q053","Chandrayaan-3 took 40 days from launch to landing. Mangalyaan took 323 days from launch to Mars orbit insertion. How many more days did Mangalyaan’s journey take than Chandrayaan-3’s?",{"kind":52,"answer":788,"tolerance":282,"unit":718},283,[790],"Subtract the two mission durations: 323 − 40.","323 − 40 = **283 days** — Mars is vastly further away than the Moon, so even with efficient orbit-raising, the journey is much longer.",[723,793],"combining facts",{"id":795,"section":35,"level":49,"prompt":796,"check":797,"hints":808,"solution":810,"skills":811},"gravity.q054","Why do astronauts aboard the ISS float?",{"kind":67,"options":798,"correct":807},[799,801,803,805],{"id":70,"label":800},"There is no gravity that far from Earth",{"id":73,"label":802},"They are falling continuously, along with the station and everything in it",{"id":76,"label":804},"The station spins to cancel out gravity",{"id":79,"label":806},"They are too far away to be pulled by anything",[73],[809],"Gravity at 400 km is still very strong — think about what \"falling together\" would feel like.","**(b).** Gravity at ISS altitude is about 89 % of its strength on the ground. Astronauts float because the station, and everyone and everything inside it, are all falling at exactly the same rate, so nothing presses on anything.",[35,812],"misconception",{"id":814,"section":35,"level":49,"prompt":815,"check":816,"hints":823,"solution":825,"skills":826},"gravity.q055","\"There is no gravity in space\" — is this statement about the ISS true or false?",{"kind":67,"options":817,"correct":822},[818,820],{"id":70,"label":819},"True",{"id":73,"label":821},"False",[73],[824],"Gravity has infinite reach and follows an inverse-square law — it never actually reaches zero.","**False.** At 400 km up, gravity is still about 8.7 N\u002Fkg, roughly 89 % of its strength at the ground. If gravity really vanished there, the station would fly off in a straight line instead of orbiting.",[35,827],"true\u002Ffalse",{"id":829,"section":35,"level":109,"prompt":830,"check":831,"hints":834,"solution":836,"skills":837},"gravity.q056","Gravity at the ISS’s altitude is about 8.69 N\u002Fkg, compared with about 9.82 N\u002Fkg at Earth’s surface. Roughly what percentage of the surface value is that? Round to the nearest whole percent.",{"kind":52,"answer":832,"tolerance":5,"unit":833},88.5,"%",[835],"Divide 8.69 by 9.82 and multiply by 100.","8.69 ÷ 9.82 × 100 ≈ **88.5 %**, close to the commonly quoted figure of about 89 %.",[838,839],"percentages","ISS gravity",{"id":841,"section":35,"level":109,"prompt":842,"check":843,"hints":854,"solution":856,"skills":857},"gravity.q057","In which of these situations would you genuinely be weightless, even if only briefly?",{"kind":67,"options":844,"correct":853},[845,847,849,851],{"id":70,"label":846},"Standing still on the ground",{"id":73,"label":848},"Sitting in a cruising aeroplane at constant height",{"id":76,"label":850},"The instant after jumping, before you land",{"id":79,"label":852},"Riding an elevator moving up at constant speed",[76],[855],"Weightlessness needs you to be in free fall — nothing holding you up or slowing you down.","**(c).** From the moment your feet leave the ground to the moment they touch again, only gravity is acting on you — genuine, if brief, free fall. The other three all have a surface (ground, seat, floor) pushing back on you.",[193,368],{"id":859,"section":35,"level":148,"prompt":860,"check":861,"hints":863,"solution":865,"skills":866},"gravity.q058","An 80 kg astronaut weighs 784.0 N on Earth’s surface (g = 9.8 N\u002Fkg). Using g = 8.69 N\u002Fkg at ISS altitude, what is her weight there, in newtons?",{"kind":52,"answer":862,"tolerance":5,"unit":55},695.5,[864],"weight = mass × g, using the ISS value of g.","weight = 80 × 8.69 = **695.5 N** — noticeably less than on the ground, but far from zero.",[61,839],{"id":868,"section":35,"level":148,"prompt":869,"check":870,"hints":872,"solution":874,"skills":875},"gravity.q059","Using the 784.0 N (Earth) and 695.5 N (ISS) weights of the same 80 kg astronaut, by what percentage has her weight decreased? Round to one decimal place.",{"kind":52,"answer":871,"tolerance":591,"unit":833},11.3,[873],"Percentage decrease = (difference ÷ original) × 100.","Difference = 784.0 − 695.5 = 88.5 N. Percentage decrease = 88.5 ÷ 784.0 × 100 ≈ **11.3 %**.",[876],"percentage change",{"id":878,"section":35,"level":173,"prompt":879,"check":880,"hints":891,"solution":893,"skills":894},"gravity.q060","A lift cable snaps (purely hypothetically) and the lift falls freely. A ball inside is released in mid-air at the same instant. What does someone standing inside the lift see the ball do?",{"kind":67,"options":881,"correct":890},[882,884,886,888],{"id":70,"label":883},"It falls to the floor quickly, faster than the lift",{"id":73,"label":885},"It hangs in mid-air, appearing not to fall at all, relative to the lift",{"id":76,"label":887},"It shoots upward, away from the person",{"id":79,"label":889},"It falls, but more slowly than the person",[73],[892],"The lift, the person and the ball are all in free fall together — think about what \"falling at the same rate\" looks like from inside.","**(b).** The ball, the person and the lift are all accelerating downward at exactly g, so relative to each other, nothing appears to move. This is Einstein’s equivalence-principle thought experiment: inside a sealed, windowless, freely falling lift, you could not tell you were falling at all.",[895,368],"equivalence principle",{"id":897,"section":39,"level":49,"prompt":898,"check":899,"hints":910,"solution":912,"skills":913},"gravity.q061","What does the period of a simple pendulum (swinging through a small angle) depend on?",{"kind":67,"options":900,"correct":909},[901,903,905,907],{"id":70,"label":902},"Only its length",{"id":73,"label":904},"Only the mass of the bob",{"id":76,"label":906},"Its length and the local value of g",{"id":79,"label":908},"Its length and its mass, equally",[76],[911],"T = 2π√(L\u002Fg) — look at what appears in the formula.","**(c).** T = 2π√(L ÷ g): only length and g appear. The mass of the bob does not affect the period at all, which surprises most people the first time they test it.",[39,914],"T=2π√(L\u002Fg)",{"id":916,"section":39,"level":109,"prompt":917,"check":918,"hints":920,"solution":922,"skills":923},"gravity.q062","Using T = 2π√(L\u002Fg) with g = 9.8 N\u002Fkg, find the period of a 0.4 m pendulum, in seconds, to three decimal places.",{"kind":52,"answer":919,"tolerance":450,"unit":271},1.269,[921],"T = 2 × π × √(0.4 ÷ 9.8).","T = 2π√(0.4 ÷ 9.8) = 2π√0.0408 ≈ **1.269 s**.",[914],{"id":925,"section":39,"level":109,"prompt":926,"check":927,"hints":929,"solution":931,"skills":932},"gravity.q063","A 0.6 m pendulum is set swinging. Using T = 2π√(L\u002Fg), how long would 20 complete swings take, in seconds, to one decimal place?",{"kind":52,"answer":928,"tolerance":591,"unit":271},31.1,[930],"Find the period of one swing first, then multiply by 20.","T = 2π√(0.6 ÷ 9.8) ≈ 1.555 s. For 20 swings: 1.555 × 20 ≈ **31.1 s**.",[914,933],"scaling up",{"id":935,"section":39,"level":148,"prompt":936,"check":937,"hints":939,"solution":942,"skills":943},"gravity.q064","A class times 20 swings of a 0.6 m pendulum at 31.1 seconds. Using g = 4π²L ÷ T², what value of g (in N\u002Fkg) does this give? Round to one decimal place.",{"kind":52,"answer":938,"tolerance":54,"unit":512},9.8,[940,941],"Find the period of one swing: 31.1 ÷ 20.","g = 4 × π² × 0.6 ÷ period².","Period = 31.1 ÷ 20 = 1.555 s. g = 4π² × 0.6 ÷ 1.555² ≈ **9.8 N\u002Fkg** — a close match to the accepted 9.8.",[944,945],"g=4π²L\u002FT²","measuring g",{"id":947,"section":39,"level":148,"prompt":948,"check":949,"hints":951,"solution":953,"skills":954},"gravity.q065","On Mars, g = 3.72 N\u002Fkg. Using T = 2π√(L\u002Fg), what would the period of a 1.0 m pendulum be there, in seconds, to two decimal places?",{"kind":52,"answer":950,"tolerance":622,"unit":271},3.26,[952],"T = 2π√(1.0 ÷ 3.72).","T = 2π√(1.0 ÷ 3.72) ≈ **3.26 s** — much longer than the roughly 2.01 s the same pendulum takes on Earth, because Mars’s gravity is weaker.",[914,62],{"id":956,"section":39,"level":173,"prompt":957,"check":958,"hints":960,"solution":963,"skills":964},"gravity.q066","A different class uses a 1.2 m pendulum and times 20 swings at 44.2 seconds. Using g = 4π²L ÷ T², what value of g, in N\u002Fkg, does their experiment give? Round to one decimal place.",{"kind":52,"answer":959,"tolerance":54,"unit":512},9.7,[961,962],"Find the period per swing first: 44.2 ÷ 20.","g = 4 × π² × 1.2 ÷ period².","Period = 44.2 ÷ 20 = 2.21 s. g = 4π² × 1.2 ÷ 2.21² ≈ **9.7 N\u002Fkg**, close to the accepted 9.8, with the small gap explained by ordinary experimental error.",[944,965],"experimental error",{"id":967,"section":43,"level":109,"prompt":968,"check":969,"hints":980,"solution":982,"skills":983},"gravity.q067","What did Galileo’s ramp experiments show that Aristotle’s ideas had missed?",{"kind":67,"options":970,"correct":979},[971,973,975,977],{"id":70,"label":972},"That heavier objects always fall faster",{"id":73,"label":974},"That falling distance grows with the square of time, and mass does not change the rate",{"id":76,"label":976},"That objects fall at a constant speed",{"id":79,"label":978},"That gravity does not exist",[73],[981],"Aristotle’s idea was based on everyday observation in air, without separating out the effect of air resistance.","**(b).** By timing balls rolling down gentle ramps, Galileo found that distance grows with the square of the time, and that mass makes no difference to the rate of falling — directly contradicting Aristotle’s \"heavier falls faster\" idea.",[984,985],"history","Galileo",{"id":987,"section":43,"level":109,"prompt":988,"check":989,"hints":993,"solution":995,"skills":996},"gravity.q068","What one-word name is given to the near-weightless environment experienced by astronauts in continuous free fall (not \"zero gravity\")?",{"kind":990,"accept":991},"text",[992],"microgravity",[994],"It is not really \"zero\" gravity — think of a word suggesting a very small amount.","The correct term is **microgravity** — tiny residual effects, not a true absence of gravity. Astronauts and their surroundings are all falling at the same rate.",[997,35],"vocabulary",{"id":999,"section":43,"level":148,"prompt":1000,"check":1001,"hints":1012,"solution":1014,"skills":1015},"gravity.q069","In general relativity, what is gravity best described as?",{"kind":67,"options":1002,"correct":1011},[1003,1005,1007,1009],{"id":70,"label":1004},"A pulling force that acts instantly across any distance",{"id":73,"label":1006},"The curving of spacetime by mass and energy, with objects following the straightest available path",{"id":76,"label":1008},"A kind of magnetism",{"id":79,"label":1010},"A force that only works between planets, not everyday objects",[73],[1013],"Newton described gravity as a force; Einstein’s description is different.","**(b).** Einstein’s general relativity treats gravity not as a pulling force but as the effect of mass and energy curving spacetime, with objects (including light) simply following the straightest path through that curve.",[1016],"general relativity",{"id":1018,"section":43,"level":173,"prompt":1019,"check":1020,"hints":1023,"solution":1026,"skills":1027},"gravity.q070","Using r = 2GM ÷ c², with G = 6.674 × 10⁻¹¹ N·m²\u002Fkg², the Sun’s mass M = 1.9885 × 10³⁰ kg, and c = 2.998 × 10⁸ m\u002Fs, calculate the Sun’s Schwarzschild radius, in kilometres, to two decimal places.",{"kind":52,"answer":1021,"tolerance":152,"unit":1022},2.95,"km",[1024,1025],"Compute 2GM first, then divide by c², then convert metres to kilometres.","2 × 6.674 × 10⁻¹¹ × 1.9885 × 10³⁰ ≈ 2.654 × 10²⁰.","r = (2 × 6.674 × 10⁻¹¹ × 1.9885 × 10³⁰) ÷ (2.998 × 10⁸)² ≈ 2,953 m ≈ **2.95 km**. The Sun would have to be crushed to this size, with no mass lost, to become a black hole.",[1028,1029],"Schwarzschild radius","black holes",[1031,1032,1033,1034,1035,1036,1037,1038],"gravity-nasa-planetary-factsheet","gravity-hyperphysics-gravity","gravity-physicsclassroom-free-fall","gravity-physicsclassroom-universal-gravitation","gravity-nasa-iss","gravity-isro-chandrayaan3","gravity-wiki-mars-orbiter-mission","gravity-britannica-gravity","needs_review",{"generatedBy":1041,"notes":1042},"claude-code","Draft. All 70 numeric answers computed and asserted in Python (common.py functions), matching the values used in the five layers. Pending owner review.","7727dfd37224aa12c844c66aa3829304f90925ea13e5b0e6bfc5769775775f23",{"logic:questions":1045,"source:gravity-britannica-gravity":1046,"source:gravity-hyperphysics-gravity":1047,"source:gravity-isro-chandrayaan3":1048,"source:gravity-nasa-iss":1049,"source:gravity-nasa-planetary-factsheet":1050,"source:gravity-physicsclassroom-free-fall":1051,"source:gravity-physicsclassroom-universal-gravitation":1052,"source:gravity-wiki-mars-orbiter-mission":1053},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","8a68b69eee24a181ce32e96e29a3f9d05221ee983d659a93152038805c82ae06","ad0b66634225092960d8f417463e2d74017822199cc2886992381487d6b8cbd9","4a3fd27e9999dffc0827fe18aa9f0049368cf20460f5ac72888566b7373d5a10","bfc3220795160e3b0ddf890b11835e5452f06f2c29b03d7541b29110035b675a","9c57a129761cacac5b2946c90eff33acba841871c7bbd0b69bdd1ec642680db9","8c311b8ddd919ef66d07a631e86b5823f4c525e1b8bf33ae94ffd5f066a7d201","a331b2d122b65b125c1004d73ef7cb3806c57d8dcb358b67a4f6cc16473bfe0f","a2bd9540c2f98b549ba5a3f8a01d306df9c3db545b3d8572d93f8f6bed5c6998","preview-7e1cbbcc4f",1789899599686]