An astronaut of mass 82.2 kg is 31.2 m far out in space, with both the space ship and the astronaut at rest with respect to each other. Without a thruster, the only way to return to the ship is to throw his 0.502 kg wrench directly away from the ship. If he throws the wrench with a speed of 20.0 m/s, how many seconds does it take him to reach the ship?
0 =m1•v1 – m2•v2,
v1 =m2•v1/m1 =0.502•20/82.2 =0.122 m/s.
t = s/v1 =31.2/0.122 = 255.7 s =4.26 min
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces act on it.
First, let's consider the system of the astronaut and the wrench. Before the wrench is thrown, the total momentum of the system is zero because both the astronaut and the wrench are at rest. Therefore, the initial momentum of the system is zero.
After the wrench is thrown, the total momentum of the system remains zero. But now, the wrench is moving away from the astronaut, so the astronaut must have a momentum in the opposite direction to maintain a zero total momentum.
The momentum of an object is given by the product of its mass and velocity. Let's denote the velocity of the astronaut after throwing the wrench as "V" and the velocity of the wrench as "v".
The initial momentum of the system is zero, so we have:
(0.502 kg)(20.0 m/s) + (82.2 kg)(0 m/s) = (0.502 kg)(v) + (82.2 kg)(-v)
Simplifying the equation, we get:
10.04 kg·m/s = (0.502 kg - 82.2 kg)(v)
Now, we can solve for "v":
10.04 kg·m/s = (-81.698 kg)(v)
v ≈ -0.123 m/s
The negative sign indicates that the wrench is moving in the opposite direction of the astronaut.
Now, to find the time it takes for the astronaut to reach the ship, we can use the equation for velocity:
v = d/t
Rearranging the equation, we have:
t = d/v
Substituting the values, we get:
t = 31.2 m / (-0.123 m/s)
t ≈ -254.63 s
The negative sign means that the time is negative, which is not physically meaningful. Therefore, we disregard the negative sign and take the positive value.
Therefore, it takes the astronaut approximately 254.63 seconds to reach the ship.