Mini Physics
Part of Mini EducationSwitch subject
SearchBookmarksMenu

Energy dissipation and rebounds

Key idea: Friction, air resistance and collisions transfer mechanical energy to internal stores and sound. With no energy input, dissipation leaves less kinetic and gravitational energy for a rebound.

Reading settings

On this page

Reading settings

Text size, line spacing and reading width for your lessons.

Text size
Line spacing
Reading width

Live preview

Take a moment to read at your own pace. Choose the text size and spacing that feel comfortable for you.

Pause after each idea, then explain it in your own words before moving on.

Narrower lines take effect when the screen has room.

Enlarged diagram

Without a continuing energy supply, resistance can slow moving objects down. A sledge loses mechanical energy while sliding, and an ordinary ball released onto a stationary floor bounces lower each time. The energy is not destroyed: friction, air resistance and collisions transfer energy from kinetic and gravitational stores into internal stores of the object and its surroundings. Adding one term to the energy account tells you how much is left.

Add the dissipated energy to the account

Sliding friction dissipates mechanical energy into the internal stores of contacting surfaces. Air resistance also dissipates energy when an object moves relative to the air. In the falling and sliding cases here, the object, surfaces and air warm up slightly. Some energy is also carried away by sound. This energy is dissipated: it still exists, but it is spread through the surroundings and no longer helps the motion.

For the motion considered here, with no continuing energy input and with only kinetic, gravitational and dissipated energy to account for:

E_(p,i) + E_(k,i) = E_(p,f) + E_(k,f) + E_dissipated

The dissipated term goes on the right because it is energy that left the kinetic and gravitational stores during the motion. Add it to the final stores and the total matches the start, so energy is still conserved.

  1. Sketch the start and end positions, note the speeds and choose the level where h = 0.
  2. Calculate the total KE and GPE at the start.
  3. Subtract the dissipated energy to find the total KE and GPE at the end.
  4. Use whichever store is unknown to find a speed or a height.

Worked example 1

Block on a rough slope

Problem

A 5.0 kg block slides from rest down a rough slope through a vertical height of 5.0 m. Friction dissipates 30 J on the way down. Neglect air resistance and take g = 10 N kg⁻¹. Find the block’s kinetic energy at the bottom.

Show full solution

  1. Calculate the starting energy

    Method

    Take the bottom of the slope as h = 0 and find the GPE at the top.

    Reason

    The block starts from rest, so all its starting energy is in the gravitational store.

    Working

    Answer

    E_(p,i) = mgh = (5.0)(10)(5.0) = 250 J

  2. Write the account with the dissipated energy

    Method

    At the bottom, the GPE is zero; the rest is KE and dissipated energy.

    Reason

    Friction transferred 30 J from the kinetic and gravitational stores to the internal stores of the block and slope.

    Working

    Answer

    250 J = E_(k,f) + 30 J

  3. Find the kinetic energy

    Working

    Answer

    E_(k,f) = 250-30 = 220 J

Guided practice 2

Dropping a ball with air resistance

About 6 min

Problem

A 0.500 kg ball is dropped from rest through 10 m. During the fall, 10 J is dissipated by air resistance. Take g = 10 N kg⁻¹. Find its speed just before it hits the ground.

Subtract the loss before finding speed

Show the steps Optional working
Unit: J
Unit: J
Unit: m/s

Hints

Hint 1

find the starting energy

Which energy store contains the ball’s energy at release?

Hint 2

include air resistance

Subtract 10 J before using Eₖ = (1/2)mv².

Show solution step by step

  1. Find the starting GPE

    Method

    Calculate the energy at release, taking the ground as h = 0.

    Reason

    The ball starts from rest 10 m above the ground.

    Working

    Answer

    Eₚ = (0.500)(10)(10) = 50 J

  2. Account for dissipation

    Method

    Subtract the energy transferred by air resistance.

    Reason

    The 10 J is now in the internal stores of the air and ball, so only 40 J reaches the ball’s kinetic store.

    Working

    Answer

    Eₖ = 50-10 = 40 J

  3. Calculate the speed

    Reason

    Only energies add and subtract. Kinetic energy depends on speed squared, so convert the remaining 40 J to a speed only at the end.

    Working

    Answer

    v = square root of (2Eₖ/m) = square root of (2(40)/0.500) = square root of 160 = 12.6 m s⁻¹ ≈ 13 m s⁻¹

Check your understanding 1 Maximum height with air resistance

A 0.20 kg ball is thrown vertically upwards from the ground at 10 m/s. Air resistance dissipates 4.0 J on the way up. Take g = 10 N/kg. Find the initial kinetic energy, the GPE at the highest point and the maximum height.

Show the steps Optional working
Unit: J
Unit: J
Unit: m
Show answer

Initial KE is (1/2)(0.20)(10)² = 10 J. At the top KE is zero, so 10 = mgh + 4.0 and GPE is 6.0 J. Thus h = 6.0/(0.20 × 10) = 3.0 m.

Without resistance it would reach 5.0 m. Compare the energy account before using mgh to find height.

A bouncing ball

A ball dropped onto a stationary floor rebounds to a lower height because energy is dissipated. Follow its energy from release to the top of the first bounce, taking the ground as h = 0 and neglecting air resistance.

  1. At release, the ball is at rest at height hᵢₙᵢₜᵢₐₗ. All its energy is GPE, mghᵢₙᵢₜᵢₐₗ.
  2. Just before impact, the ball is at ground level, so its GPE is zero. All the GPE has become KE.
  3. During impact, the ball squashes and its KE is stored briefly in its elastic store. As it springs back into shape, most of this energy returns to the kinetic store. Some is transferred to the internal stores of the ball and the ground, and some is carried away by sound.
  4. At the top of the bounce, the ball is momentarily at rest at height hₘₐₓ. All the remaining energy is GPE, mghₘₐₓ.

Because some energy was dissipated in the impact, mghₘₐₓ is less than mghᵢₙᵢₜᵢₐₗ, so hₘₐₓ < hᵢₙᵢₜᵢₐₗ. The fraction of energy kept decides the height of the bounce: a ball that keeps 60% of its energy rises to 60% of its starting height.

Energy accounting through a bounceFive equal-length bars show an illustrative 100 J initial energy. It is all GPE at release, all KE before impact, 80 J elastic and 20 J dissipated at maximum compression, 60 J KE and 40 J dissipated after rebound, and 60 J GPE and 40 J dissipated at the rebound top. Elastic energy is temporarily stored during contact; further dissipation occurs while the ball recovers. Grey segments show dissipation to internal stores and sound. Air resistance and the small height change during contact are neglected; these values are illustrative, not measured.A bounce temporarily stores elastic energyIllustrative 100 J model; air resistance and contact-height changes neglected.1 Released from rest100 J GPE2 Just before impact100 J KE3 Maximum compressionvertical speed momentarily zero80 J elastic20 J4 Just after rebound60 J KE40 J5 Rebound top60 J GPE40 JGrey: energy dissipated to internal storesand by sound
Energy accounting through a bounceFive equal-length bars show an illustrative 100 J initial energy. It is all GPE at release, all KE before impact, 80 J elastic and 20 J dissipated at maximum compression, 60 J KE and 40 J dissipated after rebound, and 60 J GPE and 40 J dissipated at the rebound top. Elastic energy is temporarily stored during contact; further dissipation occurs while the ball recovers. Grey segments show dissipation to internal stores and sound. Air resistance and the small height change during contact are neglected; these values are illustrative, not measured.A bounce temporarily stores elastic energyIllustrative 100 J model; air resistance and contact-height changes neglected.1 Released from rest100 J GPE2 Just before impact100 J KE3 Maximum compressionvertical speed momentarily zero80 J elastic20 J4 Just after rebound60 J KE40 J5 Rebound top60 J GPE40 JGrey: energy dissipated to internal stores and by sound
Each illustrative bar accounts for the original 100 J. Grey tracks energy dissipated to internal stores and sound, leaving less energy for the rebound.

If air resistance is not negligible, energy is also dissipated during each flight, not only in the impact.

Check your understanding 2 Energy dissipated in a bounce

A 0.500 kg ball is thrown vertically downwards at 5.0 m/s from 10 m above the ground. It rebounds to 8.0 m. Take g = 10 N/kg and ignore air resistance. Calculate the energy dissipated in the bounce.

Show the steps Optional working
Unit: J
Unit: J
Unit: J
Unit: J
Show answer

Initial energy is (1/2)(0.500)(5.0)² + (0.500)(10)(10) = 6.25 + 50 = 56.25 J.

At the rebound top KE is zero and GPE is (0.500)(10)(8.0) = 40 J. Dissipated energy is 56.25-40 = 16.25 J, or 16 J to two significant figures.

Compare your complete account. Treating the ball as dropped would omit its initial 6.25 J KE and give only 10 J dissipated.

Check your understanding 3 Separate flight losses from the bounce

A 1.0 kg ball is released from rest 2.0 m above the floor. Take g = 10 N/kg. The downward flight dissipates 2.0 J, the bounce keeps 60% of the impact KE, and the upward flight dissipates 1.0 J. Find the rebound height.

Show the steps Optional working
Unit: J
Unit: J
Unit: J
Unit: m
Show answer

Start with mgh = 20 J. Just before impact KE is 20-2.0 = 18 J. Just after the bounce it is 0.60 × 18 = 10.8 J. The upward flight leaves 10.8-1.0 = 9.8 J GPE, so

h = 9.8/(1.0 × 10) = 0.98 m.

Compare each stage with your working. Taking 60% of the original height would give 1.2 m and omit both flight losses. The bounce percentage applies to the energy entering the collision.

Common mistakes

  • Leaving out the initial KE of an object that is thrown rather than released from rest.
  • Subtracting the dissipated energy from the speed instead of from the energy.
  • Putting the dissipated energy on the wrong side of the account, which makes the final energy larger than the starting energy.
  • Assuming that all the dissipation happens in the bounce when air resistance acts during the flight too.
  • Assuming that a frictionless surface also means no air resistance. State each effect you neglect.

Spotted an error? Report a correction

Syllabus and review details
Back to top