A 95 kg astronaut and a 1500 kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of 0.15 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 from the shuttle to the astronaut?
To solve this problem, we will use the principles of conservation of momentum and the fact that the net external force acting on the system is zero.
First, let's determine the initial momentum of the system before the astronaut pushes the satellite. We can use the equation:
Initial Momentum = Final Momentum
The initial momentum is the sum of the momentum of the astronaut (ma) and the satellite (ms). Since both of them are at rest initially, their momenta are zero:
Initial Momentum = ma * 0 + ms * 0
Initial Momentum = 0
Next, let's determine the final momentum of the system after the astronaut pushes the satellite. The satellite acquires a speed of 0.15 m/s away from the shuttle. Using the equation for momentum:
Final Momentum = ma * va + ms * vs
where ma and ms are the masses of the astronaut and satellite, and va and vs are their final velocities, respectively.
Since the astronaut pushes the satellite directly away from the shuttle, the final velocity of the astronaut (va) is also 0.15 m/s. The final velocity of the satellite (vs) is 0 since it is now in contact with the shuttle.
Final Momentum = ma * 0.15 + ms * 0
Final Momentum = 0.15 * ma
Since momentum is conserved, the initial and final momenta are equal:
Initial Momentum = Final Momentum
0 = 0.15 * ma
Solving for ma, we find:
ma = 0 / 0.15
ma = 0
Since the mass of the astronaut is zero, it means that the astronaut must be weightless or not present. Therefore, the initial distance from the shuttle to the astronaut is undefined because there is no astronaut in this scenario.