An astronaut of mass 84.0 kg is taking a space walk to work on the International Space Station. Because of a malfunction with the booster rockets on his spacesuit, he finds himself drifting away from the station with a constant speed of 0.570 m/s. With the booster rockets no longer working, the only way for him to return to the station is to throw the 7.85 kg wrench he is holding.
In which direction should he throw the wrench?
toward the station away from the station Correct: Your answer is correct.
He throws the wrench with speed 16.67 m/s WITH RESPECT TO HIMSELF.
After he throws the wrench, how fast is the astronaut drifting toward the space station?
Incorrect: Your answer is incorrect.
Your response differs from the correct answer by more than 10%. Double check your calculations.
What is the speed of the wrench with respect to the space station?
To find the speed of the astronaut drifting toward the space station after throwing the wrench, we can use the principle of conservation of momentum.
The initial momentum of the astronaut before throwing the wrench is given by:
Mass of astronaut (m1) * Initial velocity of astronaut (v1)
The momentum of the wrench after being thrown is given by:
Mass of wrench (m2) * Velocity of wrench (v2)
Since there is no external force acting on the system, the total momentum before throwing the wrench should be equal to the total momentum after throwing the wrench.
Mathematically, it can be written as:
(m1 * v1) + (m2 * 0) = (m1 * v1 + m2 * v2)
where v1 is the initial velocity of the astronaut (0.570 m/s) and m2 is the mass of the wrench (7.85 kg).
We need to find v2, the velocity of the wrench relative to the space station.
Simplifying the equation:
m1 * v1 = (m1 * v1) + (m2 * v2)
Rearranging the equation:
m1 * v1 - m1 * v1 = m2 * v2
0 = m2 * v2
Since the velocity of the astronaut relative to the space station is zero, the velocity of the wrench relative to the space station (v2) is also zero.
Therefore, the speed of the wrench with respect to the space station is 0 m/s.
To determine the speed of the astronaut drifting toward the space station after throwing the wrench, we can use the principle of conservation of momentum.
First, let's find the momentum of the astronaut before he throws the wrench. The equation for momentum is given by:
momentum = mass * velocity
The mass of the astronaut is 84.0 kg, and his velocity is given as 0.570 m/s. So, the initial momentum of the astronaut is:
momentum_astronaut = 84.0 kg * 0.570 m/s
Now, let's consider the wrench. The astronaut throws the wrench in the opposite direction he wants to move, so the momentum of the wrench will have a negative sign. The mass of the wrench is given as 7.85 kg, and its velocity with respect to the astronaut is 16.67 m/s. So, the momentum of the wrench is:
momentum_wrench = -7.85 kg * 16.67 m/s
According to the principle of conservation of momentum, the sum of the initial momentum of the astronaut and the wrench should be equal to the final momentum after the throw. Since there are no external forces acting on the system, the total momentum is conserved.
Therefore, we can write the equation:
momentum_astronaut + momentum_wrench = final momentum
Substituting the values we have, we get:
(84.0 kg * 0.570 m/s) + (-7.85 kg * 16.67 m/s) = final momentum
Simplifying the equation, we find:
final momentum = momentum_astronaut + momentum_wrench
final momentum = (84.0 kg * 0.570 m/s) - (7.85 kg * 16.67 m/s)
Calculating the final momentum, we get:
final momentum = 47.88 kg·m/s - 130.8395 kg·m/s
Therefore, the final momentum is:
final momentum = -82.9595 kg·m/s
Since momentum is a vector quantity (it has both magnitude and direction), the negative sign indicates that the astronaut is moving in the opposite direction of the wrench's momentum. So, the astronaut is drifting toward the space station with a speed of 82.9595 kg·m/s.