Elastic Potential Energy Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Elastic Potential Energy.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Energy stored in an elastic object that has been stretched or compressed from its natural length.

A stretched rubber band 'wants' to snap back—that desire is stored energy.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Elastic Potential Energy asks what energy enters, leaves, stays stored, or changes form in the chosen system.

Common stuck point: Students often know a formula related to elastic potential energy but skip the recognition step: Can I define the system and track energy before and after the interaction or process? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Can I define the system and track energy before and after the interaction or process?

Worked Examples

Example 1

easy
A spring with spring constant k=400 N/mk = 400 \text{ N/m} is compressed by 0.05 m0.05 \text{ m}. How much elastic potential energy is stored in the spring?

Answer

PEelastic=0.5 JPE_{\text{elastic}} = 0.5 \text{ J}

First step

1
Use the elastic potential energy formula: PEelastic=12kx2PE_{\text{elastic}} = \frac{1}{2}kx^2.

Full solution

  1. 2
    PE=12(400)(0.05)2=12(400)(0.0025)=0.5 JPE = \frac{1}{2}(400)(0.05)^2 = \frac{1}{2}(400)(0.0025) = 0.5 \text{ J}
  2. 3
    This energy is available to be converted to kinetic energy when the spring is released.
Elastic potential energy is stored in deformed elastic objects such as springs, rubber bands, and bows. It depends on the square of the deformation, so doubling the compression quadruples the stored energy.

Example 2

medium
A toy dart gun has a spring (k=250 N/mk = 250 \text{ N/m}) compressed by 0.08 m0.08 \text{ m}. If the dart has a mass of 0.01 kg0.01 \text{ kg}, what speed does the dart have when it leaves the gun?

Example 3

medium
A spring (k=200k = 200 N/m) is compressed 0.10.1 m and launches a 0.250.25 kg ball horizontally on a frictionless surface. Find the launch speed.

Example 4

medium
Two springs, k1=200k_1 = 200 N/m and k2=600k_2 = 600 N/m, are each stretched by 0.10.1 m. Find the total elastic PE.

Example 5

hard
A 22 kg mass on a vertical spring (k=800k = 800 N/m) compresses it 0.050.05 m at equilibrium. It is pushed down an additional 0.100.10 m and released. Using energy methods, find the speed of the mass as it passes through the equilibrium position.

Example 6

hard
Two identical springs (k=400k = 400 N/m each) are connected in series and stretched a total of 0.20.2 m. Find total stored elastic PE.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A bungee cord (k=50 N/mk = 50 \text{ N/m}) stretches 4 m4 \text{ m} beyond its natural length when a jumper reaches the lowest point. How much elastic PE is stored?

Example 2

hard
A 0.2 kg0.2 \text{ kg} ball is launched vertically by a spring (k=500 N/mk = 500 \text{ N/m}) compressed by 0.1 m0.1 \text{ m}. How high does the ball rise above the release point? Use g=9.8 m/s2g = 9.8 \text{ m/s}^2.

Example 3

easy
A spring (k = 200 N/m) is stretched 0.3 m. Find its elastic PE.

Example 4

easy
A spring (k = 50 N/m) is compressed 0.2 m. Find its elastic PE.

Example 5

easy
In PE = \frac{1}{2}kx^2, what is x measured from?

Example 6

easy
If a spring's stretch doubles, by what factor does its elastic PE change?

Example 7

easy
A spring stores 2 J at x = 0.1 m. Find k.

Example 8

easy
Elastic PE is measured in what unit?

Example 9

easy
Two springs stretched 0.1 m: k = 100 and k = 300 N/m. Which stores more elastic PE?

Example 10

easy
A spring at its natural length (x = 0). What is its elastic PE?

Example 11

medium
A spring (k = 400 N/m) is compressed 0.15 m and launches a 0.5 kg ball horizontally (frictionless). Find the launch speed.

Example 12

medium
A spring stores 18 J when stretched 0.3 m. How much does it store when stretched 0.6 m (same k)?

Example 13

medium
A 2 kg mass on a spring (k = 800 N/m) is pulled 0.1 m and released on a frictionless surface. Find its max speed.

Example 14

medium
How much work is required to stretch a spring (k = 500 N/m) from 0.1 m to 0.2 m?

Example 15

medium
A spring is compressed and stores 12 J. If k = 600 N/m, how far was it compressed?

Example 16

medium
A bow stores 60 J of elastic PE and transfers it to a 0.05 kg arrow (no loss). Find the arrow's launch speed.

Example 17

medium
Spring A (k = 100, x = 0.4 m) and spring B (k = 400, x = 0.2 m). Which stores more elastic PE?

Example 18

challenge
A 0.3 kg ball is pressed onto a vertical spring (k = 1200 N/m), compressing it 0.1 m, then released (g = 9.8). How high above the release point does the ball rise?

Example 19

challenge
A spring (k = 250 N/m) is stretched 0.4 m. A 1 kg block attached to it on a frictionless surface is released. Find its speed when the stretch is 0.2 m.

Example 20

challenge
Two identical springs (each k = 300 N/m) are connected in parallel and stretched 0.2 m together. Find the total elastic PE stored.

Example 21

medium
A spring is stretched so it stores 50 J at x = 0.5 m. How much does it store at x = 0.1 m (same k)?

Example 22

medium
A spring (k = 1000 N/m) is compressed 0.06 m. Find the elastic PE stored.

Example 23

easy
A spring with k=100k = 100 N/m is stretched 0.20.2 m. Find the stored elastic PE.

Example 24

easy
If a spring's compression triples, by what factor does its stored elastic PE change?

Example 25

easy
A spring stores 4.54.5 J when stretched 0.30.3 m. Find kk.

Example 26

easy
A spring (k=600k = 600 N/m) is compressed 0.050.05 m. Find PEPE.

Example 27

medium
A spring stores 1616 J at x=0.4x = 0.4 m. Find the stored PE at x=0.1x = 0.1 m (same spring).

Example 28

medium
A spring (k=800k = 800 N/m) is stretched from x1=0.05x_1 = 0.05 m to x2=0.15x_2 = 0.15 m. Find the work needed.

Example 29

medium
A 0.40.4 kg block on a frictionless surface compresses a spring (k=500k = 500 N/m) by 0.20.2 m, then is released. Find the block's speed at the natural length.

Example 30

medium
A spring stores 99 J at x=0.3x = 0.3 m. Find the spring constant and the force needed to hold it there.

Example 31

medium
A toy gun spring (k=400k = 400 N/m) stores enough energy to give a 0.020.02 kg dart a speed of 1010 m/s. Find the compression xx.

Example 32

medium
A spring (k=150k = 150 N/m) is stretched 0.40.4 m. How much work was done on the spring?

Example 33

hard
A spring stores 3030 J of elastic PE when stretched 0.20.2 m. A 1.51.5 kg block attached on a frictionless surface is released. Find the block's speed when the stretch is 0.10.1 m.

Example 34

hard
A spring gun (k=1000k = 1000 N/m, compression 0.080.08 m) launches a 0.050.05 kg ball vertically (assume PE lost in spring height change is negligible). Use g=9.8g = 9.8 m/s2^2. Find maximum height above launch.

Example 35

hard
A 0.60.6 kg block slides on a frictionless surface at 44 m/s into a spring (k=600k = 600 N/m). Find the maximum compression.

Example 36

hard
A trampoline can be modeled as k=4000k = 4000 N/m. A 5050 kg child stretches it 0.30.3 m at the bottom of a jump. Find the elastic PE stored.

Example 37

hard
Two springs in parallel each with k=250k = 250 N/m are stretched 0.10.1 m by the same block. Find the total elastic PE.

Example 38

hard
A spring (k=250k = 250 N/m) is stretched from x1=0.10x_1 = 0.10 m to x2=0.20x_2 = 0.20 m. Find the work done on the spring.

Example 39

challenge
A spring (k=500k = 500 N/m) launches a 0.20.2 kg ball up a 30°30° frictionless incline from compression x=0.15x = 0.15 m. Find the distance dd the ball travels along the incline. Use g=9.8g = 9.8 m/s2^2.

Example 40

challenge
A 0.50.5 kg block sliding at 66 m/s hits a spring (k=900k = 900 N/m). A constant friction force of 33 N acts during compression. Find the maximum compression.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

potential energyspring force