A 96-kg astronaut and a 1700-kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of 0.16 m/s directly away from the shuttle. Seven-and-a-half seconds later the astronaut comes into contact with the shuttle. What was the initial distance (meters) from the shuttle to the astronaut?
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If we assume the Shuttle is an inertial reference system (it's not, because the Shuttle, astronaut, and satellite are moving in an orbit around the Earth, which requires an acceleration toward the Earth, but the error in the assumption is negligible for this problem), then the astronaut's momentum relative to the Shuttle, given by pa0 = ma * va0, where ma is the mass of the astronaut and va0 is the initial velocity of the astronaut, is pa0 = 96 kg * 0 m/s = 0 kg m/s. Note that we're using velocity, which is a vector, so momentum is a vector. (But since the magnitude is zero, we don't have a direction). The satellites momentum relative to the Shuttle is ps0 = ms * vs0 = 1700 kg * 0 m/s = 0 kg m/s. The total momentum is the vector sum of pa0 and ps0, pa0 + ps0 = 0 kg m/s.
As long as there's no outside force acting on the astronaut-satellite system (again, we're neglecting gravity, since we're assuming the Shuttle is an inertial system), the total momentum of the system must always be zero. So, if the astronaut pushes the satellite and gives it a speed of 0.16 m/s directly away from the Shuttle (which we'll call the positive direction), then the satellite has a momentum of
ps1 = 1700 kg * 0.16 m/s = 272 kg m/s
Now, the total momentum after the push is pa1 + ps1, and that must be equal to the total momentum before the push, pa0 + ps0 = 0 kg m/s. So,
pa1 + ps1 = 0 kg m/s
pa1 = -ps1
pa1 = -272 kg m/s (remember, negative is toward the Shuttle)
So that's the momentum of the astronaut. Now, we can find the velocity after the push.
pa1 = ma * va1
va1 = pa1 / ma
va1 = -272 kg m/s / 96 kg
va1 = -2.83 m/s (negative is toward the Shuttle)
Now, if we assume that the force was impulsive (a very short duration, basically infinite acceleration), we can figure out how far the astronaut was from the Shuttle. That's just a distance-rate-time problem. The distance traveled by the astronaut in time t is
d = va1 * t
d = -2.83 m/s * 7.5 s
d = -21.25 m (negative is toward the Shuttle)
So, the astronaut traveled 21.25 m toward the Shuttle in 7.5 s after pushing the satellite. I hope that helps!
As long as there's no outside force acting on the astronaut-satellite system (again, we're neglecting gravity, since we're assuming the Shuttle is an inertial system), the total momentum of the system must always be zero. So, if the astronaut pushes the satellite and gives it a speed of 0.16 m/s directly away from the Shuttle (which we'll call the positive direction), then the satellite has a momentum of
ps1 = 1700 kg * 0.16 m/s = 272 kg m/s
Now, the total momentum after the push is pa1 + ps1, and that must be equal to the total momentum before the push, pa0 + ps0 = 0 kg m/s. So,
pa1 + ps1 = 0 kg m/s
pa1 = -ps1
pa1 = -272 kg m/s (remember, negative is toward the Shuttle)
So that's the momentum of the astronaut. Now, we can find the velocity after the push.
pa1 = ma * va1
va1 = pa1 / ma
va1 = -272 kg m/s / 96 kg
va1 = -2.83 m/s (negative is toward the Shuttle)
Now, if we assume that the force was impulsive (a very short duration, basically infinite acceleration), we can figure out how far the astronaut was from the Shuttle. That's just a distance-rate-time problem. The distance traveled by the astronaut in time t is
d = va1 * t
d = -2.83 m/s * 7.5 s
d = -21.25 m (negative is toward the Shuttle)
So, the astronaut traveled 21.25 m toward the Shuttle in 7.5 s after pushing the satellite. I hope that helps!