Mechanical 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 Mechanical 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

The total of kinetic energy and potential energy in a mechanical system at any given moment.

The combined 'useful' energy for mechanical motion — kinetic plus all forms of potential 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: Mechanical 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 mechanical 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

1 \text{ kg}

ball at height

5 \text{ m}

moves at

6 \text{ m/s}

. What is its total mechanical energy? Use

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

Answer

E = 68 J

First step

KE:

\frac{1}{2}mv^2 = \frac{1}{2}(1)(36) = 18 \text{ J}

Full solution

2
PE: $mgh = 1 \times 10 \times 5 = 50 \text{ J}$ .
3
Total: $E = K E + P E = 18 + 50 = 68 J$

Mechanical energy is the sum of kinetic and potential energy. In the absence of friction and other non-conservative forces, it remains constant.

Example 2

medium

2 \text{ kg}

object slides down a frictionless ramp from

8 \text{ m}

high. What is its speed at the bottom? Use

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

Example 3

medium

1 \text{ kg}

ball is dropped from

10 \text{ m}

(

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

, no air resistance). Using mechanical-energy conservation, find its speed when it is

4 \text{ m}

above the ground.

Example 4

medium

A pendulum bob swings from rest at

0.2 \text{ m}

above its lowest point. Find its maximum speed (

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

Example 5

hard

A roller coaster car of mass

400 \text{ kg}

at the top of a

25 \text{ m}

hill moves at

5 \text{ m/s}

(

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

, no friction). Find its speed at a low point

5 \text{ m}

above the ground.

Practice Problems

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

Example 1

medium

A ball with

68 \text{ J}

of mechanical energy is at height

3 \text{ m}

. Its mass is

1 \text{ kg}

. What is its speed? Use

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

Example 2

easy

4 \text{ kg}

projectile has

200 \text{ J}

of kinetic energy at a height of

5 \text{ m}

. What is its mechanical energy? Use

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

Example 3

easy

A 2 kg ball moves at 3 m/s and sits 5 m above the ground. Find its mechanical energy (use

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

Example 4

easy

A 4 kg object has

KE = 50 \text{ J}

and

PE = 30 \text{ J}

. What is its mechanical energy?

Example 5

easy

A stationary 3 kg book rests 2 m above the floor (use

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

). What is its mechanical energy?

Example 6

easy

A 1 kg puck slides on level ice at 4 m/s with zero height. Find its mechanical energy.

Example 7

easy

If mechanical energy is conserved, a pendulum has

80 \text{ J}

at the top where it is momentarily at rest. What is its

KE

at the lowest point?

Example 8

easy

A spring stores

12 \text{ J}

of elastic PE and a block on it moves with

7 \text{ J}

of KE. What is the total mechanical energy?

Example 9

easy

A 5 kg rock is dropped from rest. The instant before release, what is its kinetic energy?

Example 10

easy

A system has

ME = 100 \text{ J}

and

PE = 40 \text{ J}

. What is its kinetic energy?

Example 11

medium

A 2 kg ball is thrown up at 10 m/s from the ground (use

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

, no air resistance). Find the maximum height using mechanical energy.

Example 12

medium

A 0.5 kg ball is dropped from 20 m (use

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

, no air resistance). Find its speed just before hitting the ground.

Example 13

medium

A 3 kg cart rolls down a frictionless ramp from 4 m high (use

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

). What is its KE at the bottom?

Example 14

medium

A 1 kg ball at 6 m/s rises to where its speed is 2 m/s. How much PE did it gain? (no friction)

Example 15

medium

A 2 kg block slides down a rough incline from 5 m high (use

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

) and arrives at the bottom with

KE = 70 \text{ J}

. How much energy was lost to friction?

Example 16

medium

A pendulum bob of mass 0.2 kg swings from a height of 0.45 m (use

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

, no friction). Find its speed at the lowest point.

Example 17

medium

A 10 kg object has mechanical energy 500 J. If its PE is 200 J, how fast is it moving?

Example 18

medium

A roller-coaster car (frictionless) starts at rest 30 m high (use

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

). How fast is it going at a point 10 m high?

Example 19

challenge

A 1 kg block slides down a frictionless ramp from 2 m, then onto a rough flat surface (

\mu = 0.25

) where it stops. How far does it travel on the flat part? (use

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

)

Example 20

challenge

A 2 kg ball is launched up at 20 m/s. Air resistance dissipates 40 J before it reaches the top. Find the maximum height (use

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

Example 21

challenge

On a frictionless track, a 0.5 kg cart at the bottom moving at 8 m/s must clear a loop. It reaches the top of a 2 m loop. What is its speed at the top? (use

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

)

Example 22

medium

A 1 kg ball is thrown up at 12 m/s (use

g = 10

, no air resistance). Find its speed when it is 5 m high.

Example 23

easy

An object has

KE = 25 \text{ J}

and

PE = 75 \text{ J}

. What is its mechanical energy?

Example 24

easy

2 \text{ kg}

ball is at rest at height

4 \text{ m}

(

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

). Find its mechanical energy.

Example 25

easy

0.4 \text{ kg}

cart moves at

5 \text{ m/s}

on the ground (height = 0). Find its mechanical energy.

Example 26

easy

A pendulum has

ME = 12 \text{ J}

and at one instant its kinetic energy is

7 \text{ J}

. Find the potential energy.

Example 27

easy

A spring is compressed and stores

30 \text{ J}

of elastic PE. A

2 \text{ kg}

cart attached to it is at rest at

h=0

. Find the total mechanical energy of the system.

Example 28

medium

0.5 \text{ kg}

ball thrown straight up reaches a max height of

5 \text{ m}

(

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

, no air resistance). What was its launch speed?

Example 29

medium

3 \text{ kg}

object at

2 \text{ m}

height moves at

4 \text{ m/s}

(

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

). Find its mechanical energy.

Example 30

medium

2 \text{ kg}

block slides down a frictionless incline from

3 \text{ m}

(

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

). Find its speed at the bottom.

Example 31

medium

5 \text{ kg}

object slides on a rough surface and loses

40 \text{ J}

of mechanical energy to friction over

2 \text{ m}

. Find the friction force.

Example 32

medium

A spring of constant

k = 200 \text{ N/m}

is compressed by

0.2 \text{ m}

on a frictionless surface and launches a

0.5 \text{ kg}

block. Find the block's launch speed.

Example 33

medium

2 \text{ kg}

block slides on a frictionless surface at

3 \text{ m/s}

into a spring with

k = 50 \text{ N/m}

. How far does the spring compress at maximum?

Example 34

medium

1.5 \text{ kg}

ball is dropped from

20 \text{ m}

and lands with speed

18 \text{ m/s}

(

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

). How much energy was lost to air resistance?

Example 35

hard

0.2 \text{ kg}

ball is launched at

15 \text{ m/s}

horizontally from a

5 \text{ m}

cliff (

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

, no air resistance). Find its speed just before hitting the ground.

Example 36

hard

0.4 \text{ kg}

ball slides down a frictionless ramp from

1.6 \text{ m}

and then runs onto a rough patch where

\mu_k = 0.4

(

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

). How far on the patch does it travel before stopping?

Example 37

hard

0.5 \text{ kg}

ball on a

1 \text{ m}

string swings in a vertical circle. At the lowest point its speed is

8 \text{ m/s}

(

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

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

Example 38

hard

50 \text{ kg}

skier starts at rest at the top of a

40 \text{ m}

hill (

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

). At the bottom the skier moves at

20 \text{ m/s}

. How much energy did friction and air drag dissipate?

Example 39

hard

0.1 \text{ kg}

ball is dropped onto a vertical spring (

k = 200 \text{ N/m}

) from a height

1 \text{ m}

above the spring's top (

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

, no air resistance). Find the maximum compression of the spring (ignore PE change during compression).

Example 40

hard

2 \text{ kg}

block slides down a

30^\circ

frictionless incline of length

4 \text{ m}

starting from rest (

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

). Find its speed at the bottom.

Example 41

challenge

0.5 \text{ kg}

block slides on a frictionless track from rest at height

H

, around a vertical loop of radius

R = 1 \text{ m}

. Find the minimum

H

so the block just maintains contact at the top of the loop (

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

Example 42

challenge

0.3 \text{ kg}

block is pushed against a spring (

k = 600 \text{ N/m}

) compressed by

0.1 \text{ m}

on a horizontal surface, then released. The block slides

1.5 \text{ m}

before stopping. Find the coefficient of kinetic friction (

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

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Related Concepts

Kinetic Energy Potential Energy Conservation of Energy

Background Knowledge

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

kinetic energypotential energy