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Problem 18 Medium Difficulty

At a time when mining asteroids has become feasible, astronauts have connected a line between their $3500-\mathrm{kg}$ space tug and a $6200-\mathrm{kg}$ asteroid. Using their tug's engine, they pull on the asteroid with a force of 490 $\mathrm{N}$ . Initially the tug and the asteroid are at rest, 450 $\mathrm{m}$ apart. How much time does it take for the tug and the asteroid to meet?

Answer

64$s$

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Video Transcript

according to Newton Start Law. That tug is actually being pulled by the asteroid. We force off 490 new terms. Now we can use Newton's second law on each off them separately to calculate acceleration off them separately. So for the asteroid, we have the following. The Net force acting on the asteroid is equal to its mess, finds its acceleration. Let's call it a for acceleration off the asteroid. Then the net force is the coast in 490 noodles. The net force acting on the asteroid on Lee and these these equals two seeks to double zero times acceleration off the asteroid. Then the acceleration off the asteroid is the coast to 490 divided by 62 double zeros. There is a simplification that can be done here, and then we have that acceleration off. The asteroid is a close to 49 divided by 620 which is approximately 0.8 Reader spur second squared, then the position off the asteroid as a function of time. Is it cost, too? Its initial position. Plus it's initial velocity times the time, plus its acceleration times, the time squared, divided by truth. But note that its initial position, we can say, is it close to zero. So here is Europe. Its initial velocity is the cost of zero because it meetings at rest. So the only term that is left is the last one. So for the asteroid, we have the following the asteroid position as a function of time. Is it cost too? 0.8 times T squared, divided by truth which results in 0.4 times t squared. And only that Now let me organize this information here and do the same thing for the tow. Nice. Now we repeat this calculation for the tug. So for the time you have the following the net force acting on the tug is he goes to its mass times its acceleration that we call it 80 then Oh, okay. One more thing. Let me define everything that is pointing to the right as things never that are pointing towards the positive direction on everything that is pointing to the left will be pointing to the negative direction as a consequence. Then the net force that is acting on the tug is he goes to minus 490 because it's pointing to the negative direction and its Sequels. True. 35. The zeros, the mass off the tile times acceleration off the hook, Then acceleration off the Turk is because two minus for 90 divided by 35 double zeros. That is a simplification, and this gives us an acceleration off approximately minus 0.14 meters per second squared. Now we can calculate the position off the tug as a function of time as follows. The position of the child has a function off time. Is he close to initial position? Plus, it's initial velocity times time plus its acceleration times Time squared, divided by two. Its initial position is a question for 150. Its initial velocity easy, close to zero. Then the position of the drug is a function of time. Easy, close to 450 minus. Because the acceleration is negative. 0.14. He's weird, divided by truth, and we can solve the second fraction to get the following 450 minus 0.0 of 70 squared. No, Maybe you're the nice to board again. Fine. So now we have to solve the meeting, so when they meet they you'll be at the same position. So when they meet as a will, the cost to SD So 0.0 for T squared we'll be close to 450 minus 0.0 70 squared. Now we have to sew for heat. We can do this by sending this term to the other side first. So we have 0.0 40 squared plus 0.0 70 square being because to 450 then zero point eleventy squared is equal to 450 show he squared is equal to 450 divided by 0.11 and then he is He goes to the square it off 450 divided by 0.11 which is approximately 64 seconds