To find the final speed of the crate, we need to apply the concept of work and energy. The work done on an object is equal to the change in its kinetic energy.
1. Let's calculate the work done by the astronaut while pushing the crate forward:
Work done = force x distance
Work done = 240 N x 4 m
Work done = 960 Joules
2. The work done by the astronaut increases the kinetic energy of the crate. So, the increase in kinetic energy is equal to the work done:
ΔKE = 960 J
3. We can use the equation for kinetic energy to calculate the initial kinetic energy of the crate:
KE = 0.5 x mass x velocity^2
0.5 x 110 kg x (2 m/s)^2 = 220 Joules
4. The final kinetic energy of the crate will be the sum of the initial kinetic energy and the increase in kinetic energy:
Final KE = Initial KE + ΔKE
Final KE = 220 J + 960 J
Final KE = 1180 Joules
5. To find the final speed, we need to rearrange the kinetic energy equation:
Final KE = 0.5 x mass x final velocity^2
1180 J = 0.5 x 110 kg x final velocity^2
Divide both sides by (0.5 x 110 kg):
final velocity^2 = 1180 J / (0.5 x 110 kg)
final velocity^2 = 21.45 m^2/s^2
6. Finally, take the square root of both sides to find the final velocity:
final velocity = √(21.45 m^2/s^2) ≈ 4.63 m/s
Therefore, the final speed of the crate is approximately 4.63 m/s.