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 .

Using the equation  we can find the distance at which the ball was dropped. Notice that the mass of the ball does not matter in this problem. We are told that the ball is dropped from rest making, , thus we have . When we plug in our values, and assuming that acceleration is equal to gravity (10m/s²) we find that   = 125m.

Since the object is dropped, the inital velocity is zero. Gravity is the only acceleration, the time is ten seconds, and the distance at which the object travels is unknown.

The equation  can be used to find the distance traveled.

Using the equation , we can solve for time.

Since the object started at rest, . Now we are left with the equation .

Plugging in the remaining values we can find that t = 5.5s.

Use the following kinematic equation: .

We can choose the ground to be the zero distance so that and .

Also, the initial speed is zero.

The kinematic equation simplifies using these values.

Solve to isolate the time variable.

We know that the acceleration is the acceleration due to gravity. Now we can plug in the known values and solve.