Conservation of Energy Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conservation of 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

A fundamental law of physics stating that the total energy of an isolated system remains constant over time — energy can be transferred between objects.

Energy is like money—you can spend it, save it, or change its form, but you can't make more out of nothing.

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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: Conservation of 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 conservation of 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

medium

2 \text{ kg}

ball is dropped from

20 \text{ m}

. What is its speed just before hitting the ground? Use

g = 10 \text{ m/s}^2

Answer

v = 20 m/s

First step

At the top:

PE = mgh = 2 \times 10 \times 20 = 400 \text{ J}

KE = 0

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Example 2

hard

A roller coaster car (

500 \text{ kg}

) starts from rest at

30 \text{ m}

high and descends to

10 \text{ m}

. What is its speed at

10 \text{ m}

? Use

g = 10 \text{ m/s}^2

Example 3

medium

0.2\text{ kg}

ball is thrown straight up at

12\text{ m/s}

. Find the maximum height it reaches. Use

g = 10\text{ m/s}^2

Example 4

medium

0.3\text{ kg}

block on a frictionless surface is hit by a spring (PE

= 6\text{ J}

) and slides up a frictionless ramp. How high does it rise? Use

g = 10\text{ m/s}^2

Example 5

hard

A block slides down a frictionless ramp of height

3\text{ m}

, then across a rough horizontal

5\text{ m}

stretch with

\mu_k = 0.3

. Find its speed at the end. Use

g = 10\text{ m/s}^2

and

m = 2\text{ kg}

Practice Problems

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

Example 1

medium

A pendulum of length

2 \text{ m}

is released from a height

0.5 \text{ m}

above its lowest point. What is its speed at the lowest point? Use

g = 10 \text{ m/s}^2

Example 2

hard

A skier (

60 \text{ kg}

) starts from rest at the top of a

25 \text{ m}

hill and reaches the bottom at

18 \text{ m/s}

. How much energy was lost to friction? Use

g = 10 \text{ m/s}^2

Example 3

easy

A 1 kg ball falls from 5 m (g = 9.8). Find its KE at the bottom (frictionless).

Example 4

easy

Is total energy conserved in an isolated system?

Example 5

easy

A pendulum at its highest point has 6 J of PE and 0 KE. What is its KE at the lowest point (frictionless)?

Example 6

easy

A frictionless system has 40 J total mechanical energy. If KE = 25 J, what is PE?

Example 7

easy

If friction is present, is mechanical energy alone conserved?

Example 8

easy

A 2 kg object at rest at 3 m falls (g = 9.8). What is its total mechanical energy throughout (frictionless)?

Example 9

easy

When a ball bounces lower each time, where did the missing energy go?

Example 10

easy

A roller coaster is highest at the start. Where is its speed greatest (frictionless)?

Example 11

medium

A 0.5 kg ball is thrown up at 8 m/s (g = 9.8). Find its KE when it is 2 m high.

Example 12

medium

A 3 kg cart at 4 m/s rolls up a frictionless hill (g = 9.8). How high does it rise?

Example 13

medium

A 2 kg block slides down a frictionless 5 m high slope, then up to 3 m on the other side. What is its KE at the 3 m point (g = 9.8)?

Example 14

medium

A 1 kg ball dropped from 4 m onto a spring (k = 500 N/m), g = 9.8. Find the spring's max compression (assume small compared to drop, ignore extra PE in spring).

Example 15

medium

A 0.2 kg ball is launched at 10 m/s up a frictionless ramp. Find its speed when it has risen 2 m (g = 9.8).

Example 16

medium

A 2 kg block at 6 m/s slides onto a rough patch and stops. How much energy was dissipated as heat?

Example 17

medium

A 0.1 kg ball dropped from 2 m bounces back to 1.5 m (g = 9.8). How much energy was lost in the bounce?

Example 18

challenge

A 0.5 kg block compresses a spring (k = 200 N/m) by 0.3 m on a frictionless surface, is released, then slides up a ramp. How high does it rise (g = 9.8)?

Example 19

challenge

A 4 kg block slides 5 m down a 3 m high frictionless slope, then crosses a rough flat 2 m stretch with friction force 10 N (g = 9.8). Find its speed after the rough patch.

Example 20

challenge

A pendulum of length 1 m is released from horizontal (g = 9.8). Find the speed of the bob at the lowest point.

Example 21

medium

A 1 kg ball at 5 m/s at ground level rolls up a frictionless ramp (g = 9.8). Find its speed when it has risen 0.5 m.

Example 22

medium

A 0.5 kg ball dropped from 3 m rebounds to 2.4 m (g = 9.8). What fraction of its mechanical energy was retained?

Example 23

easy

0.5\text{ kg}

ball is dropped from

10\text{ m}

. Find its speed just before hitting the ground (frictionless). Use

g = 10\text{ m/s}^2

Example 24

easy

A pendulum has

20\text{ J}

of PE at its highest point and

0

KE. What is its KE at the lowest point (frictionless)?

Example 25

easy

1\text{ kg}

object has

50\text{ J}

of mechanical energy. If its PE is

30\text{ J}

, what is its KE?

Example 26

easy

2\text{ kg}

object is at rest

4\text{ m}

above the ground. Find its initial PE. Use

g = 10\text{ m/s}^2

Example 27

easy

A roller coaster rises and falls without friction. At point A it has

100\text{ J}

PE and

0\text{ J}

KE; at point B it has

40\text{ J}

PE. What is its KE at B?

Example 28

medium

3\text{ kg}

cart at

2\text{ m/s}

rolls up a frictionless ramp. How high does it rise? Use

g = 10\text{ m/s}^2

Example 29

medium

50\text{ kg}

skier descends a

20\text{ m}

hill and arrives at the bottom at

15\text{ m/s}

. How much mechanical energy was lost to friction? Use

g = 10\text{ m/s}^2

Example 30

medium

0.5\text{ kg}

block slides down a frictionless ramp of height

1.8\text{ m}

, then onto level ground. Find its speed on the ground. Use

g = 10\text{ m/s}^2

Example 31

medium

A pendulum of length

0.8\text{ m}

is released from

40^\circ

from vertical. Find the speed at the bottom. Use

g = 10\text{ m/s}^2

Example 32

medium

A spring with

k = 100\text{ N/m}

is compressed

0.2\text{ m}

, then released to launch a

0.4\text{ kg}

block on a frictionless surface. Find the block's speed when the spring relaxes.

Example 33

medium

1.5\text{ kg}

ball is dropped from

5\text{ m}

and bounces back to

4\text{ m}

. How much energy was lost in the bounce? Use

g = 10\text{ m/s}^2

Example 34

medium

0.4\text{ kg}

ball is dropped from rest at

2\text{ m}

and hits a spring with

k = 200\text{ N/m}

. Ignoring the spring's height, find the maximum spring compression. Use

g = 10\text{ m/s}^2

Example 35

medium

2\text{ kg}

block slides

4\text{ m}

along a horizontal surface with

\mu_k = 0.25

before stopping. Find its initial speed. Use

g = 10\text{ m/s}^2

Example 36

hard

0.1\text{ kg}

ball at

30\text{ m/s}

encounters air drag that does

25\text{ J}

of work over its trajectory. Find its KE at the end.

Example 37

hard

A roller coaster car (

800\text{ kg}

) descends a frictionless hill from rest at

40\text{ m}

and enters a vertical loop of radius

10\text{ m}

. Find its speed at the top of the loop. Use

g = 10\text{ m/s}^2

Example 38

hard

0.05\text{ kg}

bullet at

200\text{ m/s}

embeds in a

1.95\text{ kg}

block initially at rest on a frictionless surface. After the inelastic collision they compress a spring (

k = 400\text{ N/m}

). Find the max compression.

Example 39

hard

1\text{ kg}

block slides down a

5\text{ m}

ramp at

30^\circ

with

\mu_k = 0.2

. Find its speed at the bottom. Use

g = 10\text{ m/s}^2

Example 40

hard

A pump lifts

100\text{ kg}

of water per minute to a height of

15\text{ m}

. Find the pump's minimum power. Use

g = 10\text{ m/s}^2

Example 41

challenge

0.5\text{ kg}

ball is released from rest at height

h

above a vertical loop of radius

0.4\text{ m}

. Find the minimum

h

so the ball maintains contact at the top of the loop. Use

g = 10\text{ m/s}^2

Example 42

challenge

0.2\text{ kg}

ball moving at

5\text{ m/s}

hits a spring on a frictionless surface and compresses it. The spring constant is

k = 50\text{ N/m}

. Find the maximum compression.

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Related Concepts

Kinetic Energy Potential Energy Work-Energy Theorem Thermal Energy

Background Knowledge

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

kinetic energypotential energy