A hovercraft of mass 77.0 kg can move on a horizontal surface, the x-y plane. A single unbalanced force acts on the hovercraft, but the size of the force is unknown. The hovercraft initially has a velocity of 6.0 m/s in the positive x direction and some time later has a velocity of 9.0 m/s in the positive y direction. How much work is done on the hovercraft by the force during this time?

On second thought, the work done equals the kinetic energy change. You can solve for that, no matter how long the velocity change takes. The direction changes don't matter in the result.

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To calculate the work done on the hovercraft by the force during the time interval, we need to determine the change in kinetic energy.

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. The formula for work done is given by:

Work = Change in Kinetic Energy

Kinetic Energy (KE) = 1/2 * mass * velocity^2

Let's calculate the initial kinetic energy (KE1) and final kinetic energy (KE2):

KE1 = 1/2 * mass * velocity1^2
= 1/2 * 77.0 kg * (6.0 m/s)^2
= 1/2 * 77.0 kg * 36.0 m^2/s^2
= 1386.0 Joules

KE2 = 1/2 * mass * velocity2^2
= 1/2 * 77.0 kg * (9.0 m/s)^2
= 1/2 * 77.0 kg * 81.0 m^2/s^2
= 2971.5 Joules

Now, we can find the change in kinetic energy (ΔKE) by subtracting the initial kinetic energy from the final kinetic energy:

ΔKE = KE2 - KE1
= 2971.5 Joules - 1386.0 Joules
= 1585.5 Joules

Therefore, the work done on the hovercraft by the force during this time interval is 1585.5 Joules.

How long does the velocity change take? "Some time later" does not provide enough information.

You can get the force from the momentum change and the time interval.