In this question, we're given a few different variables. We're told there is energy, time, displacement, force, and mass involved. So, we need to think about a way to relate these all together, given what the question is asking.

We're told that energy is being transferred to an object at a certain rate. This energy transfer results in a force on the object which, in turn, causes it to undergo displacement. Finally, we know the mass of the object as well.

Since we're told that the energy is being transferred at a certain rate, we know power is involved. We also know that the change in energy of an object is equal to the force that object experience multiplied by its distance. Thus, we can relate these variables with the following expression.

Using this expression, we can rearrange it in order to isolate the term for time.

Now, we just need to plug the values given in the question stem to solve for our answer.

Also notice that we didn't need to use the object's mass to solve this problem. That was extraneous information.

The equation for power is:

Where  is the work done on an object,  is the time it takes to do that work, and  is the power necessary to do that amount of work in that time.T

The equation for work is:

Where  is the work done on an object,  is the force applied to an object,  is the distance the object moves, and  is the angle between the applied force and the direction the object moves.

From the problem statement we have:

 and .

We can assume that the bulldozer is pushing straight in to the rock because no information is given about the angle between the force and distance, so we can assume that  and  and that part of the formula can be ignored in this problem.

Plugging in to our equation we find:

We are also given:

So we can plug in to our power equation to find:

Note: the unit for work is shown to be  and a Newton-meter is the same as a Joule . Power is measured in Watts, .

Since the car begins from rest, we have no air resistance to worry about. The question is also further simplified since we are told to ignore rotational accelerations and all friction forces. Therefore, we can simply convert our power units to a more familiar unit and go from there.

If we choose a representative time segment, we can calculate how much energy is applied toward the motion of the vehicle in that time period. From this, we can calculate the change in velocity and thus the acceleration of the car. To keep things simple, let's choose an arbitrary time segment of 1 second. Therefore, the care will have  of kinetic energy and we can use the following expression to determine its velocity:

Rearranging for velocity, we get:

Plugging in our values, we get:

Since the change in velocity was achieved over a period of 1 second, we know that the acceleration has the same numerical value:

Plugging in values:

Calculating gravitational potential energy:

Converting  to  and plugging in values:

Based on conservation of energy, this would have been the energy given to the ball by the throw.

Using

For this problem we assume that you are delivering the same amount of energy each second continuously to the light bulb, this gives us the ability to equate energy and power in this instance. We represent the relation below

We now plug in our knows, and move to isolate the rotational speed, plugging in the rotational inertia for a sphere of 

Now solving for w we arrive at

This question is testing your understanding of how to calculate the value of the power, or the rate at which work is done by a given force. In this case, we consider a box that is dropped to the ground off of a cliff, and we are told that the only force acting upon the box is gravity. Thus, we are being asked to calculate the power of gravity, which means the rate at which work is done by gravity.

The formula for power is given by, 

We can solve our problem as follows:

 (where )

Therefore, the power of gravity in this example is .