4bKinetic and Potential Energy
Explore the two forms of mechanical energy, the work-energy theorem, and the distinction between conservative and non-conservative forces.
Kinetic and potential energy are the two forms of mechanical energy — understanding them and the work-energy theorem unlocks the powerful principle of conservation of energy used throughout all of physics.
Lesson Overview
Energy is the capacity to do work. Kinetic energy (KE = ½mv²) is the energy of motion. Gravitational potential energy (PE = mgh) is stored energy due to height. Elastic potential energy (PE = ½kx²) is stored in a compressed or stretched spring. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE. Forces are classified as conservative (path-independent, like gravity) or non-conservative (path-dependent, like friction).
Key Concepts
Kinetic Energy
KE = ½mv² — depends on mass and the square of speed. SI unit: joule (J).
Gravitational PE
PE_g = mgh — depends on mass, g, and height above a reference level.
Elastic PE
PE_e = ½kx² — stored in a spring; k is spring constant (N/m), x is compression/extension.
Work-Energy Theorem
W_net = ΔKE = KE_f − KE_i. Net work equals the change in kinetic energy.
Conservative Force
Work done is independent of path; energy can be fully recovered (e.g., gravity, spring force).
Non-Conservative Force
Work done depends on path; energy is dissipated as heat (e.g., friction, air resistance).
A 2 kg ball moves at 6 m/s. What is its kinetic energy?
A 5 kg book is placed on a shelf 2 m above the floor. What is its gravitational potential energy relative to the floor?
A spring with k = 400 N/m is compressed 0.15 m. How much elastic potential energy is stored?
A net force does 200 J of work on a 4 kg object initially at rest. What is the object's final speed?
A 3 kg object moves at 4 m/s. A net force does work on it until it reaches 8 m/s. How much net work was done?
A 10 kg object is dropped from a height of 5 m. What is its kinetic energy just before hitting the ground? (Ignore air resistance.)
Hint: Use conservation of energy: the gravitational PE at the top converts entirely to KE at the bottom.
A spring (k = 200 N/m) launches a 0.5 kg ball. If the spring is compressed 0.3 m, what is the ball's speed when it leaves the spring?
Hint: Set elastic PE equal to kinetic energy: ½kx² = ½mv².
A 1500 kg car brakes from 30 m/s to rest. How much work does friction do on the car?
Hint: Use the work-energy theorem: W_net = ΔKE. The car starts at 30 m/s and ends at 0.
Is gravity a conservative or non-conservative force? Explain using the definition.
Hint: Does the work done by gravity depend on the path taken, or only on the starting and ending heights?
A 2 kg object has 50 J of kinetic energy. What is its speed?
Hint: Rearrange KE = ½mv² to solve for v.
Key Vocabulary
Kinetic Energy
The energy an object possesses due to its motion; KE = ½mv².
Example: A 1000 kg car at 20 m/s has KE = ½(1000)(400) = 200,000 J.
Potential Energy
Stored energy due to an object's position or configuration; can be gravitational (mgh) or elastic (½kx²).
Example: A stretched rubber band stores elastic potential energy.
Work-Energy Theorem
The net work done on an object equals its change in kinetic energy: W_net = ΔKE.
Example: A net force of 10 N over 5 m does 50 J of work, increasing KE by 50 J.
Conservative Force
A force for which the work done is independent of the path taken between two points.
Example: Gravity is conservative: lifting a book 1 m straight up or along a ramp requires the same work against gravity.
Interactive Practice — 5 Questions
A 4 kg object moves at 5 m/s. What is its kinetic energy?
A 3 kg object is raised 4 m. What is the increase in gravitational PE? (g = 9.8 m/s²)
The work-energy theorem states that W_net equals:
Which of the following is a non-conservative force?
A spring with k = 500 N/m is compressed 0.2 m. How much elastic PE is stored?
Independent Practice
A 0.5 kg ball is thrown upward at 12 m/s. Find (a) its initial KE, (b) its maximum height using the work-energy theorem, and (c) its KE when it returns to the starting height.
A spring (k = 300 N/m) is compressed 0.25 m and launches a 0.2 kg ball horizontally. Find the ball's launch speed.
A 1200 kg car travels at 25 m/s. The brakes apply a friction force of 6000 N. Use the work-energy theorem to find the stopping distance.
Explain why friction is classified as a non-conservative force. What happens to the energy "lost" to friction?
★ A 5 kg block slides down a frictionless ramp from a height of 3 m, then compresses a spring (k = 1000 N/m) at the bottom. Find the maximum compression of the spring. (Hint: all gravitational PE converts to elastic PE at maximum compression.)
ChallengeCommon Mistakes
Using v instead of v² in the kinetic energy formula — writing KE = ½mv.
Kinetic energy depends on the square of speed: KE = ½mv². Doubling the speed quadruples the kinetic energy.
Forgetting that gravitational PE depends on the chosen reference level.
You can choose any reference level (h = 0), but you must use it consistently throughout the problem. Only changes in PE are physically meaningful.
Applying the work-energy theorem using only one force instead of the net work.
W_net = ΔKE uses the total net work from all forces. Calculate work done by each force, then sum them to get W_net.
Math Tips
The work-energy theorem W_net = ΔKE is a powerful shortcut: if you know the net work, you can find the change in speed without knowing the details of the motion (time, acceleration). Use it whenever you need to relate force, distance, and speed.
For spring problems, remember x is the deformation from the natural (unstretched) length, not the total length of the spring. Always check whether the spring is compressed or stretched.