Kinetic & Potential Energy

You’ll be able to

Compute kinetic energy with KE = ½mv²
Compute gravitational potential energy with PE = mgh
Compute elastic potential energy stored in a spring, ½kx²

Kinetic energy: the energy of motion

Kinetic energy is the energy an object has because it is moving: KE = ½mv². Because the speed is squared, kinetic energy is very sensitive to speed — doubling the speed quadruples the kinetic energy, which is why stopping distances grow so fast with speed. Kinetic energy is always positive (or zero) and is measured in joules.

Potential energy: stored energy of position

Potential energy is energy stored because of an object’s position or configuration. Gravitational PE, PE = mgh, is the energy of height — how much work gravity could do if the object fell a height h. Only changes in height matter, so you are free to choose where h = 0. Elastic PE, ½kx², is stored in a stretched or compressed spring, where k is the spring constant and x is the distance from the spring’s natural length.

Kinetic energy

KE = ½mv²

v is speed. The ½ and the square are both essential — dropping either is a common error.

Potential energy

PE_grav = mgh · PE_spring = ½kx²

h is height above your chosen reference; x is the spring’s stretch or compression from equilibrium.

Worked example

A 3 kg ball moves at 4 m/s and is then lifted to a shelf 5 m high (g = 10 m/s²). Find its kinetic energy while moving and its gravitational potential energy on the shelf.

1.Kinetic energy: KE = ½mv² = ½ × 3 × (4)² = ½ × 3 × 16 = 24 J.
2.Gravitational PE on the shelf: PE = mgh = 3 × 10 × 5.
3.Multiply: PE = 150 J.
4.The two quantities are independent — one depends on speed, the other on height.

Answer: KE = 24 J and PE = 150 J

Watch out

Kinetic energy uses the square of speed and a factor of ½ — KE = ½mv², not mv (that is momentum). Mixing up ½mv² and mv turns an energy problem into a momentum answer.

Checkpoint

A 2 kg object moves at 6 m/s. What is its kinetic energy?

Checkpoint

A 4 kg book is lifted from the floor onto a shelf 2 m high (g = 10 m/s²). How much gravitational potential energy does it gain?

On the exam

You may place the h = 0 reference wherever it is most convenient — the floor, the tabletop, the ground below a cliff. Only the change in height between start and finish affects the physics, so choose the level that makes the arithmetic simplest.

Answer the 2 checkpoints as you read.