Kinetic and Potential Energy

What is Energy? Energy in general is a measure of the motion of objects.

There are two general categories of energy that objects may have: Kinetic energy and Potential energy.

• Kinetic energy is the energy of an object due to its motion. The faster an object is moving, the more kinetic energy it has. For example, an electron that is moving faster has more kinetic energy than a slower-moving electron.
• Potential energy is related to the amount of motion an object can have (in the future) because it is being acted on by a net force. (If you don't know what "net force" means, click here.) For example, electrons are pulled by protons toward the nucleus of atoms. Because of this, electrons have the "potential" to move even faster.

Types of Kinetic Energy

Kinetic energy is the energy of an object (including atoms and molecules!) that is due to the object’s motion. Because there are different types of motion, there are different types of kinetic energy. When people use the term "kinetic energy," they are most likely referring to the translational kinetic energy of an object. The translational kinetic energy of an object is the kinetic energy of the object due to the object as a whole moving across some distance.

• The (translational) Kinetic energy of an object is proportional to its mass and the square of its speed (or velocity). Specifically: KE = ½m*v²

In these units, if we say "kinetic energy," we mean "translational kinetic energy." If we mean another type of kinetic energy, we will say what that other kind of kinetic energy is (e.g., "rotational kinetic energy").

• Below are some examples of translational motion, where the object as a whole moves across a distance:

In each case, the (translational) kinetic energy of the object can be calculated from its mass and its speed/velocity (KE = ½m*v²).

Other types of kinetic energy

• A second type of kinetic energy is kinetic energy due to rotational motion, where an object rotates on an axis. An example of this is the Earth rotating on its axis, causing day and night. This type of motion is shown below:
• A final type of kinetic energy is kinetic energy due to vibrational motion. For this type of motion, again the object as a whole does not move through space. Instead, it moves kind of like an accordion. A person playing an accordion can stand still while playing it because it has no translational motion.

An example of vibrational motion is shown below.

In the example above, the "object" is composed of two boxes and a spring between them. The boxes both move away from each other, then back toward each other (like the accordion). But the middle of the spring (the "center of mass") does not move in space. 

Carbon dioxide is one molecule that absorbs energy from sunlight, which is converted into vibrational motion of the molecules. 

Types of kinetic energy: Understanding Check...

Q1: In the picture above, a child attaches a weight to a spring hanging from the ceiling. Then, the child pulls down on the weight. The weight moves up and down without turning. Which is true of the weight's motion?

a) The weight has both translational and vibrational motion.

b) The weight only has vibrational motion.

c) The weight only has vibrational motion.

d) The weight has neither vibrational nor rotational motion.

Submit

Right! The weight as a whole moves up and down. Therefore, it has translational motion. The weight does not vibrate (like an accordion). So, the weight only has translational motion. The answer is (c).

Well, let's think about this. The weight as a whole moves up and down. Therefore, it has translational motion. The weight does not vibrate (like an accordion). So, the weight only has translational motion. The answer is (c).

Q2: Two boys, Tom and Jerry, decide to go for a jog together. Tom has a mass of 68 kg (150 lbs). Jerry has a mass of 34 kg (75 lbs). They run at exactly the same speed. Which of the following is true of their (translational) kinetic energies while they are running?

a) Tom's kinetic energy is twice Jerry's.

b) Jerry's kinetic energy is twice Tom's.

c) Tom's kinetic energy is four times Jerry's.

d) It's impossible to know.

Submit

That's right! Kinetic energy is equal to mass times velocity squared (KE = 1/2*m*v^2.) Tom's mass is twice Jerry's mass. So, Tom's kinetic energy is twice Jerry's. (a) is the correct response.

Actually, kinetic energy is equal to mass times velocity squared (KE = 1/2*m*v^2.) Tom's mass is twice Jerry's mass. So, Tom's kinetic energy is twice Jerry's. (a) is the correct response.

((Advanced: Due to new (probably unfamiliar) terminology)) 

• Rotational kinetic energy (KE): As we said before, an object can have other types of kinetic energy besides translational KE. Rotational kinetic energy is the energy due to the object spinning around an axis (including rolling on the ground). 

• In general, rotational KE = ½I  • ω² . (This looks similar to KE = ½m • v² , right?) I is like the mass of the object, but it’s the resistance of the object’s mass to rotational motion (rather than translational motion). And ω is the rotational speed of the object or how fast the object rotates on its axis (rather than translational speed).

• If a solid ball is rolling across a surface, it will have BOTH translational KE and rotational KE. Again, the rotational KE is in general equal to: ½ I • ω².

• For a solid sphere, I = (2/5)m • r² (you need to look up values of I for different shapes of objects)

• This relationship indicates that I, the resistance to spinning, increases with both the mass (m) of the sphere and with the square of the radius of the sphere. (So, smaller spheres resist spinning less than larger spheres, which is intuitive, right?)

• ω is related to the translational speed of the sphere (if it is rolling without slipping on the surface at all) as follows: ω = v/r.

• This is because when an object rolls one revolution, or 360 degrees — which is equal to 2 π radians (trust us for now) — it will travel a distance equal to the circumference of the sphere, 2 πr (where r is the radius of the sphere) or πd(where d is the diameter of the sphere). 

• So, KE_R = (2/5) m • r² × (v/r)² → the r’s will cancel out, so we get for the rotational KE of a sphere:  KE_R = 2/5 m • v²

Q2A: How does the translational Kinetic Energy compare to the rotational Kinetic Energy (KE) of the solid sphere when it is rolling down an inclined plane?

a) The translational KE is greater than the rotational.

b) The rotational KE is greater than the translational.

c) They are the same.

d) I have no idea

Submit

Yes! The translational KE of a sphere is 1/2mv² (or 5/10mv²). The rotational KE of a sphere is 2/5mv² (or 4/10mv²). So, the translational KE is a bit (1/10) bigger. (a) is the correct response.

Not quite. The translational KE of a sphere is 1/2mv² (or 5/10mv²). The rotational KE of a sphere is 2/5mv² (or 4/10mv²). So, the translational KE is a bit (1/10) bigger. (a) is the correct response.

((End of advanced section))

Potential energy is the potential for an object to move faster (or accelerate) because a net force is acting on the object. For example, an object such as a rock at the top of a cliff (right before it starts to fall) has potential energy because the downward force of gravity will make it fall (and gain a lot of kinetic energy, if it's high enough!). 

The potential energy of an object due to gravity (when no other forces act on the object) depends on three variables: 

• the mass of the object (m)

• the acceleration due to gravity (g) (on Earth, g is about 10 m/s²), 

• and its height above the ground (h).

As each of these variables increases, the potential energy of the object increases. This is shown in the equation below:

• The potential energy of an object (due to gravity) = m x g x h

Q2.1: What is the potential energy (due to gravity) of a 0.5 kg squirrel sleeping on the ground?
Submit

There is no net force acting on the squirrel (the force of gravity down is equal to the force of the ground pushing up on the squirrel). So, the potential energy is zero.

Q3: What is the potential energy (due to gravity) of a 0.5 kg squirrel right as it starts to fall from a tree branch 10 meters above the ground?
Submit

The potential energy of the squirrel due to gravity is equal to the product of the squirrel's mass (0.5 kg), acceleration due to gravity (about 10 m/s2), and the height of the squirrel above the ground (10 m):
PE (gravity) = (.5 kg)*(10 m/s2) * (10 m) = 50 kg*m2/s2

Q4: The squirrel climbs halfway up the tree, to a branch that is 5 meters above the ground. It slips and falls off of the branch. What is its potential energy (due to gravity) right when it starts to fall?
Submit

The potential energy of the squirrel due to gravity is:
PE = mass*height*g (acceleration due to gravity, about 10 m/s^2)
So, PE = (0.5 kg)*(5 m)*(10 m/s^2) = 25 kg*m^2/s^2
This is half of the squirrel's potential energy when it was 10 meters up the tree.

• The relationship between potential and kinetic energy is discussed in more detail in the unit: Energy Can Change Forms But Must Be Conserved.

Understanding Check: Potential vs. Kinetic Energy:

Q5a: A block on a spring moves up and down (and up and down, etc.). At Time 1, the block is at its highest point. What type(s) of energy does it have at this point?

a) Only kinetic energy.

b) Both kinetic and potential energy.

c) Only potential energy.

Submit

Yes! The block is not moving at Time 1, so it has no kinetic energy. But the block has both gravity and the force of the spring pushing down on it. So it has (lots of) potential energy, or potential to move faster.

Not quite. The block is not moving at Time 1, so it has no kinetic energy. But the block has both gravity and the force of the spring pushing down on it. So it has (lots of) potential energy, or potential to move faster.

Q5b: At Time 2, the block is at its middle point. It is now moving at its maximum speed. At this point, the downward force of gravity acting on the block is equal to the upward force of the spring on the block. So, there is no net force acting on the block in the vertical (or any other!) direction. What type(s) of energy does the block have at this point?

a) Only kinetic energy.

b) Both kinetic and potential energy.

c) Only potential energy.

Submit

Good job! The block is moving at Time 2, so it has kinetic energy. Because there is no net force acting on the block at this point, the block does not have potential to move any faster. So, its potential energy at this point is zero.

Not quite. The block is moving at Time 2, so it has kinetic energy. Because there is no net force acting on the block at this point, the block does not have potential to move any faster. So, its potential energy at this point is zero.

Q5c: At Time 3, the block is at its lowest point. It is not moving at this point. There is a net force upward from the spring pulling on the block (the upward force from the spring is greater than the downward force of gravity). What type(s) of energy does it have at this point?

a) Only kinetic energy.

b) Both kinetic and potential energy.

c) Only potential energy.

Submit

Yes! The block is not moving at Time 3, so it does not have kinetic energy. However, there is a net force acting on the block, which will cause its speed to increase. So, there is potential for the block to start moving faster. We say the block now "has" potential energy.

Not quite. The block is not moving at Time 3, so it does not have kinetic energy. However, there is a net force acting on the block, which will cause its speed to increase. So, there is potential for the block to start moving faster. We say the block now "has" potential energy.