Power is equal to the Work put into the system per unit time.

Work is equal to the force acting on the object multiplied by the displacement through which it acts.

Therefore power is directly related to the force applied.

Force is also directly related to the acceleration of an object.   A constant force will create a constant acceleration.

Since power is directly related to the force applied, and the force must be constant to maintain a constant acceleration, the power must also therefore be constant.

First we need to analyze the motion of the rider on the way down the incline.  During this time, the bicyclist has a constant velocity, which means that all forces acting on him must be balanced.  He has a portion of the force of gravity pulling him down the hill and the force of friction opposing his motion.

We can use components to find the force of gravity in the x direction.

Since the net force is equal to  we know that the force of gravity in the  direction is equal to the force of friction.

Now let us consider how the bicyclist will travel when he goes up the hill.  This time, the forces acting on the bicyclist are the force of gravity and friction resisting his motion and his applied force going up the hill.

The force of friction and the force of gravity are the same as when the cyclist was coasting down the hill.

The bicyclist doesn  of work to get back up the hill.  We can now use this in our power equation to determine the amount of power used.

Power is equal to the work divided by the time to complete the work.

Work is equal to the force times the displacement through which the object moved.

We can substitute this into our power equation to get

Velocity is equal to the distance over the time.

Therefore power could be written as

Power is equal to the work divided by the time it takes to complete the work.

Work is equal to the force times the displacement.

In this case the motor is lifting the piano.  Therefore the force that it is exerting is equal to the force of gravity pulling the piano down.

We can now calculate the work done on the piano.

We can now plug this value into the power equation and solve for the time.

It will take  seconds to the lift the piano.

Power is equal to the work divided by the time to complete the work.

Work is equal to the force times the displacement through which the object moved.

We can substitute this into our power equation to get

Velocity is equal to the distance over the time.

Therefore power could be written as

First, we need to calculate the amount of energy produced by the power plant.

We know that this energy is stored in the form of gravitational potential energy.

If there is  for every  of water, we can divide this number by  to determine the number of  of water is pumped from the lower to the upper reservoir.

Known

Power = Current x Voltage

There are  in the circuit.  We can then divide this number by  to determine how many lightbulbs can go into the circuit.

 lightbulbs

First, we need to determine how much current is used by the system.

If the battery is rated at 75Ah we can divide this number by the current needed by these devices to determine how long the battery would run.

First, let us determine the amount of current in the bulb while it is in Europe.

We can use this information to determine the resistance of the bulb.

Now let us determine the current going through the bulb when it is in the United States assuming that the resistance is constant.

When we compare these two current values we can see that the bulb has a higher current in Europe and a lower current in the United States. This means that the bulb will be dimmer.

The fastest way to solve a problem like this is with momentum.

Remember that momentum is equal to mass times velocity: . We can rewrite this equation in terms of force.

Using this transformation, we can see that momentum is also equal to force times time.

  can also be thought of as .

Expand this equation to include our given values.

Since the object is not moving at the end, its final velocity is zero. Plug in the given values and solve for the force.

We would expect the answer to be negative because the force of friction acts in the direction opposite to the initial velocity.

The fastest way to solve a problem like this is with momentum.

Remember that momentum is equal to mass times velocity: . We can rewrite this equation in terms of force.

Using this transformation, we can see that momentum is also equal to force times time.

 can also be thought of as .

Expand this equation to include our given values.

Since the box is not moving at the end, its final velocity is zero. Plug in the given values and solve for the time.