A 82-kg astronaut and a 1300-kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of 0.24 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?

Remember that momentum is conserved. Momentum (P) = m*v so find P of the satelite (0.24 m/s * 1,300 kg) and then set that equal to the P of the astronaut (82kg * v). Solve for v. V should have the units m/s, then times that by 7.5 seconds, the seconds should cancel out to verify that you have your answers in meters. That is the distance from the shuttle to the astronaut. Hope that helps :)

To solve this problem, we can use the concept of conservation of momentum. The total momentum before the astronaut pushed the satellite is equal to the total momentum after.

Let's denote the initial distance from the shuttle to the astronaut as "d".

The total momentum before the astronaut pushed the satellite is the sum of the momentum of the astronaut and the momentum of the satellite. Since they are both at rest relative to the shuttle initially, their momenta are zero.

After the astronaut pushes the satellite, the momentum of the satellite is given by the equation:

Momentum = Mass x Velocity

The mass of the satellite is 1300 kg, and the velocity is 0.24 m/s. Therefore, the momentum of the satellite is:

Momentum of satellite = 1300 kg x 0.24 m/s

Now, let's denote the mass of the astronaut as "m" (82 kg).

The momentum of the astronaut can be calculated using the equation:

Momentum = Mass x Velocity

The velocity of the astronaut can be determined by dividing the distance traveled by the astronaut by the time it took to come into contact with the shuttle.

We know that the velocity of the astronaut is:

Velocity = Distance / Time

Given that the time is 7.5 seconds, we can calculate the velocity of the astronaut.

Now, we can calculate the momentum of the astronaut:

Momentum of astronaut = mass x velocity

The total momentum after the astronaut pushed the satellite is equal to the sum of the momentum of the astronaut and the momentum of the satellite.

Now, we can write the equation for conservation of momentum:

Initial momentum = Final momentum

0 (momentum of the astronaut initially) + 0 (momentum of the satellite initially) = momentum of the astronaut + momentum of the satellite

Substituting the values, we get:

0 + 0 = (mass of astronaut x velocity of astronaut) + (mass of satellite x velocity of satellite)

Since we are solving for the initial distance, we can use the relationship between distance and velocity.

Distance = Velocity x Time

Substituting the values into the equation, we get:

0 + 0 = (mass of astronaut x (Distance / Time)) + (mass of satellite x 0.24 m/s)

Simplifying the equation, we get:

0 = (mass of astronaut x Distance / Time) + (mass of satellite x 0.24 m/s)

Rearranging the equation to solve for the initial distance, we get:

Distance = -((mass of satellite x 0.24 m/s) / (mass of astronaut / Time))

Substituting the given values, we get:

Distance = -((1300 kg x 0.24 m/s) / (82 kg / 7.5 s))

Calculating this expression, we find:

Distance = - 5.6659 m

Therefore, the initial distance from the shuttle to the astronaut is approximately -5.67 meters. Note that the negative sign indicates that the astronaut was initially closer to the shuttle.