Gravitational Potential Energy - AP Physics 1
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A lazy contractor of mass 80kg who is bricking a building figured out that a seesaw makes his life much easier. He needs to deliver bricks to his coworker who is 10 meters above him. The contractor puts bricks on one side of the seesaw and jumps on the other side to fling the bricks upward. If the contractor jumps onto the seesaw from a height of 1.5 meters, what is the maximum mass of bricks he can put on the seesaw that will reach his coworker?
A lazy contractor of mass 80kg who is bricking a building figured out that a seesaw makes his life much easier. He needs to deliver bricks to his coworker who is 10 meters above him. The contractor puts bricks on one side of the seesaw and jumps on the other side to fling the bricks upward. If the contractor jumps onto the seesaw from a height of 1.5 meters, what is the maximum mass of bricks he can put on the seesaw that will reach his coworker?
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This problem can either be very simple or very complex.
You could convert the contractor's potential energy into kinetic energy, and then find the velocity at which he hits the seesaw. Then, use that to calculate the kinetic energy of the bricks and the height at which the bricks will fly. Although some may find that fun to do, it's unnecessary.
If we assume that all of the energy of the man is transferred to the bricks, we can use the conservation of energy equation:
We need to make a few clarifying statements for our initial and final states. In the initial state, the contractor is at a height of 1.5m and not moving. The bricks are not moving at this point either. In the final state, the man is on the ground not moving, and the bricks are at a height of 10m and not moving. Therefore, we can rewrite the above equation as the following:
Rearranging for the mass of the bricks:
This problem can either be very simple or very complex.
You could convert the contractor's potential energy into kinetic energy, and then find the velocity at which he hits the seesaw. Then, use that to calculate the kinetic energy of the bricks and the height at which the bricks will fly. Although some may find that fun to do, it's unnecessary.
If we assume that all of the energy of the man is transferred to the bricks, we can use the conservation of energy equation:
We need to make a few clarifying statements for our initial and final states. In the initial state, the contractor is at a height of 1.5m and not moving. The bricks are not moving at this point either. In the final state, the man is on the ground not moving, and the bricks are at a height of 10m and not moving. Therefore, we can rewrite the above equation as the following:
Rearranging for the mass of the bricks:
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A basketball player goes for a 3-point shot, but misses. The basketball has a mass of 
and is traveling at 
as it reaches its maximum height of 
. If the ball loses 15% of its total energy and half its horizontal velocity as it bounces off the rim of the hoop, what is the total velocity of the ball as it hits the ground?
A basketball player goes for a 3-point shot, but misses. The basketball has a mass of
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Using the equation for conservation of energy, we get:
Let's clarify that our initital state is when the ball is at its maximum height and the final state is when it reaches the ground. Plugging in our expressions and removing initial kinetic and final potential energy, we get:
Note that we removed initial kinetic energy despite the ball moving. At the initial state, all of the ball's velocity is horizontal. Since there is no vertical velocity, we can ignore it for now, coming back to it later.
We multiply the initial energy by 0.85 because the problem statement says that we lose 15% of the ball's total energy after hitting the rim. Rearranging for final velocity:
This is only the y-component of the final velocity. We need to combine this with the x-component to get the total velocity. The problem statement tells us that the final horizontal velocity is 
, therefore we can write:
Using the equation for conservation of energy, we get:
Let's clarify that our initital state is when the ball is at its maximum height and the final state is when it reaches the ground. Plugging in our expressions and removing initial kinetic and final potential energy, we get:
Note that we removed initial kinetic energy despite the ball moving. At the initial state, all of the ball's velocity is horizontal. Since there is no vertical velocity, we can ignore it for now, coming back to it later.
We multiply the initial energy by 0.85 because the problem statement says that we lose 15% of the ball's total energy after hitting the rim. Rearranging for final velocity:
This is only the y-component of the final velocity. We need to combine this with the x-component to get the total velocity. The problem statement tells us that the final horizontal velocity is
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Consider the following system:
If 
of work is done against the progression of the block over a distance 
, and the coefficient of kinetic friction is 
, what is the slope of the plane?
Consider the following system:
If
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This problem helps you become more comfortable with using obscure units. Since we are given work as a function of mass, we don't actually have to know the mass of the block. Furthermore, since we are neglecting air resistance, we know that all of the work is done by friction. To calculate the force of friction, we can use the following expression:
To calculate the work done by this force, we multiply by the distance that it was applied:
Since we were given work as a function of mass, we can eliminate mass to get:
Rearranging for the angle, we get:
We know all of these values, allowing us to solve for the angle:
This problem helps you become more comfortable with using obscure units. Since we are given work as a function of mass, we don't actually have to know the mass of the block. Furthermore, since we are neglecting air resistance, we know that all of the work is done by friction. To calculate the force of friction, we can use the following expression:
To calculate the work done by this force, we multiply by the distance that it was applied:
Since we were given work as a function of mass, we can eliminate mass to get:
Rearranging for the angle, we get:
We know all of these values, allowing us to solve for the angle:
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A mountain biker goes off a jump with an initial vertical velocity of 
. If the biker lands a vertical distance of 
below the launch point, what is his vertical velocity the moment he lands?
A mountain biker goes off a jump with an initial vertical velocity of
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We can use the expression for conservation of energy to solve this problem:
Assuming a final height of zero, we can eliminate final potential energy. Then, substituting in expressions for each variable, we get:
Canceling out mass and rearranging for final velocity, we get:
We can use the expression for conservation of energy to solve this problem:
Assuming a final height of zero, we can eliminate final potential energy. Then, substituting in expressions for each variable, we get:
Canceling out mass and rearranging for final velocity, we get:
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A rock is lifted 10m off of the ground, carried by the person holding it across a field of length 100m, and held at a final height of 10m. What was the change in gravitational potential energy?
A rock is lifted 10m off of the ground, carried by the person holding it across a field of length 100m, and held at a final height of 10m. What was the change in gravitational potential energy?
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The change in gravitational potential energy is only dependent on the final and initial height of the object, and is independent of the path taken to get to the final height. Because the change in height for the rock is zero, the change in gravitational potential energy equals zero.
The change in gravitational potential energy is only dependent on the final and initial height of the object, and is independent of the path taken to get to the final height. Because the change in height for the rock is zero, the change in gravitational potential energy equals zero.
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A dumbell with a mass of 
has been dropped from a height of 
. What is the amount of potential energy in the dumbell after three seconds?
A dumbell with a mass of
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Over the distance described, change in gravitational pull is negligible, so gravity can be treated as a constant. Potential energy therefore varies with the dumbell's proximity to the ground, h:
Since the dumbell has an inital position of 100m, and an initial velocity of zero, its height can be described using the kinematic equation:
From this, the potential energy at a time of three seconds is:
Over the distance described, change in gravitational pull is negligible, so gravity can be treated as a constant. Potential energy therefore varies with the dumbell's proximity to the ground, h:
Since the dumbell has an inital position of 100m, and an initial velocity of zero, its height can be described using the kinematic equation:
From this, the potential energy at a time of three seconds is:
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A ball with a mass of 2kg hangs from a cord 1.4 meters in length from the ceiling of a room with a total height of 3.1 meters. What is the gravitational potential energy of the ball relative to the ceiling? Assume the acceleration of gravity 
.
A ball with a mass of 2kg hangs from a cord 1.4 meters in length from the ceiling of a room with a total height of 3.1 meters. What is the gravitational potential energy of the ball relative to the ceiling? Assume the acceleration of gravity
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Gravitational potential energy is proportional to both the height and mass of an object. Gravitational potential energy is given by:
Gravitational potential energy is also relative to a "zero height" and in our case this is the ground. What this means is that if the ground is a our "zero height", the higher the object from the ground the greater the potential energy
Gravitational potential energy is proportional to both the height and mass of an object. Gravitational potential energy is given by:
Gravitational potential energy is also relative to a "zero height" and in our case this is the ground. What this means is that if the ground is a our "zero height", the higher the object from the ground the greater the potential energy
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Two balls of equal mass (3kg) are hanging from cords 1.5 meters and 2 meters in length from a ceiling in a room with a total height of 3 meters. Which has a greater gravitational potential energy relative to the floor? Assume the acceleration of gravity is 
.
Two balls of equal mass (3kg) are hanging from cords 1.5 meters and 2 meters in length from a ceiling in a room with a total height of 3 meters. Which has a greater gravitational potential energy relative to the floor? Assume the acceleration of gravity is
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Gravitational potential energy is proportional to both the height from a "zero potential energy" reference point (in our case the floor) and the mass. While the balls would have the same gravitational potential energy since they are the same mass, they have differing energies because of differing heights. The shortest cord, or the ball the highest from the ground will have the greatest potential energy as shown:
Gravitational potential energy is proportional to both the height from a "zero potential energy" reference point (in our case the floor) and the mass. While the balls would have the same gravitational potential energy since they are the same mass, they have differing energies because of differing heights. The shortest cord, or the ball the highest from the ground will have the greatest potential energy as shown:
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A 
ball rolls down a frictionless ramp of height 
, at the end of the ramp what will its velocity be?
A
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The first step for this problem is to determine the potential energy the ball has at the top of the ramp through this equation:
We see that it has a potential energy of 100 joules. All of this potential energy gets converted to kinetic energy as the ball falls down the ramp.
Knowing this, we can determine the velocity the ball has at the bottom of the ramp by setting the potential energy equal to the kinetic energy:
We substitute the known mass and solve for 
:
The first step for this problem is to determine the potential energy the ball has at the top of the ramp through this equation:
We see that it has a potential energy of 100 joules. All of this potential energy gets converted to kinetic energy as the ball falls down the ramp.
Knowing this, we can determine the velocity the ball has at the bottom of the ramp by setting the potential energy equal to the kinetic energy:
We substitute the known mass and solve for
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A cart is traveling at 
when it launches a ball straight into the air with initial velocity 
. Ignore air resistance.
How high will the ball go?
A cart is traveling at
How high will the ball go?
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Use conservation of energy:
Plug in known values and solve for the maximum height of the object.
Use conservation of energy:
Plug in known values and solve for the maximum height of the object.
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On the surface of planet Mars, the gravitational constant 
is 
. Considering that earth's gravitational constant is 
, how high above the surface do you need to be on planet Mars to have the same gravitational potential energy as you would if you were 
up on Earth? Assume your mass is unchanged on both planets.
On the surface of planet Mars, the gravitational constant
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Recall that the formula for gravitational potential energy is:
To determine the height on planet Mars needed to have equivalent potential energy, we can set the two equations for gravitational potential energies equal.
, where 
is mass and 
is height above the ground on Earth while 
is the height above the ground on Mars.
The mass, 
can be eliminated from both sides of the equation since they are equal. Plug in and solve for the height on Mars.
Recall that the formula for gravitational potential energy is:
To determine the height on planet Mars needed to have equivalent potential energy, we can set the two equations for gravitational potential energies equal.
The mass,
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A cat jumps to the top of a wall of height 
. Determine the cat's initial upward velocity.
A cat jumps to the top of a wall of height
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Use conservation of energy:
Since initially the cat is on the ground, and can assumed to be still at the top of the wall:
Solve for velocity:
Plugging in values
Use conservation of energy:
Since initially the cat is on the ground, and can assumed to be still at the top of the wall:
Solve for velocity:
Plugging in values
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A ball takes 
to hit the ground after being dropped from a window. Estimate the height of the window.
A ball takes
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Assume a constant acceleration we may use a kinematic formula:
is the acceleration due to gravity, 
which is 
, the initial velocity, is zero
is the unknown
is the ground, which will be a height of 
Plug in values:
Solve for 
Assume a constant acceleration we may use a kinematic formula:
Plug in values:
Solve for
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Changing which of the following variables will cause the biggest increase in gravitational potential energy?
Changing which of the following variables will cause the biggest increase in gravitational potential energy?
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Gravitational potential energy is calculated using the following equation.
Where 
is the potential energy, 
is the gravitational constant, 
is the mass of one object, 
is the mass of the other object, and 
is the distance between the two objects. Decreasing 
and increasing mass of the object by the same factor will have a similar effect on the potential energy (both will increase potential energy by the same factor); therefore, changing both of these variables by same factor will have a similar effect. Recall that the force due to gravity is
When calculating 
, 
will have the biggest effect. Don't confuse equation for potential energy with equation for force due to gravity.
Gravitational potential energy is calculated using the following equation.
Where
When calculating
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Planet A is twice as massive as Planet B. Compared to a person standing on Planet A, a person standing on Planet B will have potential energy and mass.
Planet A is twice as massive as Planet B. Compared to a person standing on Planet A, a person standing on Planet B will have potential energy and mass.
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Potential energy is calculated using the following equation.
Where 
is the potential energy, 
is the gravitational constant, 
is the mass of one object, 
is the mass of the other object, and 
is the distance between the two objects. The potential energy depends on the mass of the objects (in this case the person and the planet); therefore, the more massive planet will produce the higher potential energy.
The mass is the same regardless of the situation. Mass measures the amount of substance/elements/molecules inside a person or object's body. The composition of the person’s body won’t change (regardless of the mass of the planet); therefore, the mass will stay the same. Recall that the weight of the person, however, does change based on the planet the person is standing on. This is because the weight depends on the gravitational acceleration, which depends on the mass of the planet.
Potential energy is calculated using the following equation.
Where
The mass is the same regardless of the situation. Mass measures the amount of substance/elements/molecules inside a person or object's body. The composition of the person’s body won’t change (regardless of the mass of the planet); therefore, the mass will stay the same. Recall that the weight of the person, however, does change based on the planet the person is standing on. This is because the weight depends on the gravitational acceleration, which depends on the mass of the planet.
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What is the potential energy of a person standing on the surface of the earth?
What is the potential energy of a person standing on the surface of the earth?
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The potential energy of a person or an object on earth is calculated using the following formula.
Where 
is potential energy, 
is mass, 
is acceleration due to gravity, and 
is the height from surface of earth (sea level). The question states that the person is standing on the surface of earth; therefore, 
. This means that the person has no potential energy.
Recall that kinetic energy, on the other hand, is the energy associated with motion. It depends on the velocity of an object or person. A person on the surface of earth will have kinetic energy if the person is moving.
The potential energy of a person or an object on earth is calculated using the following formula.
Where
Recall that kinetic energy, on the other hand, is the energy associated with motion. It depends on the velocity of an object or person. A person on the surface of earth will have kinetic energy if the person is moving.
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A tennis ball of mass 
is dropped off a bridge at a height of 
over an asphalt road. The ball bounces up to a height of 
. Determine the amount of energy lost to heat.
A tennis ball of mass
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Definition of work:
Combine equations:
Substitute variables, then plug in and solve for the change in heat.
Definition of work:
Combine equations:
Substitute variables, then plug in and solve for the change in heat.
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Calculate the velocity a car would need to have at the bottom of a vertical loop of radius 12m to successfully make it through without falling off of the track.
Calculate the velocity a car would need to have at the bottom of a vertical loop of radius 12m to successfully make it through without falling off of the track.
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At the top of the loop, the centripetal acceleration will need to be at least the acceleration due to gravity:
Solve for 
The radius of the circle will be half the height:
Use conservation of energy:
Solve for 
:
Combine equations:
Plug in values:
At the top of the loop, the centripetal acceleration will need to be at least the acceleration due to gravity:
Solve for
The radius of the circle will be half the height:
Use conservation of energy:
Solve for
Combine equations:
Plug in values:
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An object is 
above the floor on a table. Estimate it's velocity upon impact with the floor.
An object is
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Using conservation of energy:
Solving for 
:
Plugging in values:
Using conservation of energy:
Solving for
Plugging in values:
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Consider the following system:
Two spherical masses, A and B, are attached to the end of a rigid rod with length l. The rod is attached to a fixed point, p, which is at a height, h, above the ground. The rod spins around the fixed point in a vertical circle that is traced in grey. 
is the angle at which the rod makes with the horizontal at any given time (
in the figure).
At what angle of 
do masses A and B have the same gravitational potential energy?
Consider the following system:
Two spherical masses, A and B, are attached to the end of a rigid rod with length l. The rod is attached to a fixed point, p, which is at a height, h, above the ground. The rod spins around the fixed point in a vertical circle that is traced in grey.
At what angle of
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We are asked to determine what orientation of the rod will result in masses A and B having the same gravitational potential energy. Therefore, we first need the equation for gravitational potential energy:
We can then write this formula for each mass:
Where the subscripts denote each mass. Now we can set the two equations equal to each other (since that is the point of the question):
Canceling out g:
We are given both masses in the problem statement, so we need to determine the height of each mass. Let's think about this situation practically; Since mass A is greater than mass B, mass B will need to be higher up for the two gravitational potential energies to be the same. Therefore, we can determine that the rod will be rotated counter clockwise compared to the original figure, and 
will be positive. With this in mind, we can create equations for the height of each mass. Let's start with mass A:
We have heigh h at which the mass is at when the rod is horizontal. As we just discussed, the mass will be slightly lower than this. How much lower? We can use the angle c and length of the rod to determine a distance d, which is how far the mass is below horizontal (imagine a right triangle between the mass, point p, and the horizontal position of mass A):
This is simply the sine function for right triangles. We can take this distance and subtract it from h to get the height of mass A:
We can do the same for mass B, except we will be adding the distance d since it will be above horizontal:
Substituting these into our reduced equation above we get:
Expanding our terms:
Getting height and angles on same sides:
Factoring each side of the equation:
Dividing so we get 
alone on one side:
Taking the inverse sine of both sides:
Finally, we have arrived to our final equation. We know every value on the right side of the equation. However, lets do a quick check and make sure our units will work out (we want a unitless value inside the parenthesis).
Now it's time to plug and chug:
We are asked to determine what orientation of the rod will result in masses A and B having the same gravitational potential energy. Therefore, we first need the equation for gravitational potential energy:
We can then write this formula for each mass:
Where the subscripts denote each mass. Now we can set the two equations equal to each other (since that is the point of the question):
Canceling out g:
We are given both masses in the problem statement, so we need to determine the height of each mass. Let's think about this situation practically; Since mass A is greater than mass B, mass B will need to be higher up for the two gravitational potential energies to be the same. Therefore, we can determine that the rod will be rotated counter clockwise compared to the original figure, and
We have heigh h at which the mass is at when the rod is horizontal. As we just discussed, the mass will be slightly lower than this. How much lower? We can use the angle c and length of the rod to determine a distance d, which is how far the mass is below horizontal (imagine a right triangle between the mass, point p, and the horizontal position of mass A):
This is simply the sine function for right triangles. We can take this distance and subtract it from h to get the height of mass A:
We can do the same for mass B, except we will be adding the distance d since it will be above horizontal:
Substituting these into our reduced equation above we get:
Expanding our terms:
Getting height and angles on same sides:
Factoring each side of the equation:
Dividing so we get
Taking the inverse sine of both sides:
Finally, we have arrived to our final equation. We know every value on the right side of the equation. However, lets do a quick check and make sure our units will work out (we want a unitless value inside the parenthesis).
Now it's time to plug and chug:
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