The kinetic energy store

Definition

Kinetic energy

Kinetic energy is energy stored by an object because it is moving.

Kinetic energy is stored by an object because it is moving, so every moving object holds some.
The size of the store depends on the object's mass and on its speed.
Speed matters more than mass, because the store depends on the square of the speed.
A stationary object has a speed of $m/s0\ \text{m/s}$ , so its kinetic energy store holds $J0\ \text{J}$ .
The store fills as an object speeds up and empties as it slows down.
Kinetic energy is measured in joules, $J\text{J}$ , like every other store.

The kinetic energy equation

The store is calculated from $\dfrac{1}{2} \times m \times v^{2}$ .
Here $K E$ is the kinetic energy in joules, $m$ is the mass in kilograms and $v$ is the speed in metres per second, $m/s\text{m/s}$ .
Square the speed before multiplying, so a speed of $m/s6.0\ \text{m/s}$ gives $m2/s2v^{2} = 36\ \text{m}^{2}/\text{s}^{2}$ .
Writing the working as $\times m \times v^{2}$ makes the factor of one half harder to forget.
Rearranging for speed gives $\sqrt{\dfrac{2 \times KE}{m}}$ , and rearranging for mass gives $\dfrac{2 \times KE}{v^{2}}$ .
Doubling the mass at the same speed doubles the store, because the mass is not squared.
Doubling the speed at the same mass makes the store four times larger, since $2^{2} = 4$ .
Tripling the speed makes the store nine times larger, since $3^{2} = 9$ .
Convert grams into kilograms, and convert any speed given in $km/h\text{km/h}$ into $m/s\text{m/s}$ , before substituting.

Kinetic energy in falling and braking

A falling object empties its gravitational potential energy store and fills its kinetic energy store, so $12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h$ when resistive forces are small.
Cancelling the mass gives $\sqrt{2 \times g \times \Delta h}$ , which shows that the landing speed does not depend on how heavy the object is.
A vehicle's kinetic energy store has to be emptied before it stops, and the work done by the braking force equals that store, so $\times d = \dfrac{1}{2} \times m \times v^{2}$ .
Because the speed is squared, doubling a car's speed makes its kinetic energy store four times larger and so needs about four times the distance to stop with the same braking force.
Those joules are transferred into the thermal energy stores of the brakes, tyres and road, which is why brake discs become hot.

Example

Kinetic energy of a cyclist

A cyclist and bicycle have a combined mass of $kg80\ \text{kg}$ and travel at $m/s6.0\ \text{m/s}$ .
The equation is $\dfrac{1}{2} \times m \times v^{2}$ .
Squaring the speed gives $v^{2} = 6.0^{2} = 36$ .
Substituting gives $\times 80 \times 36$ .
The kinetic energy store holds $1440\ \text{J}$ .

Example

A mass given in grams

A cricket ball of mass $g160\ \text{g}$ is thrown at $m/s18\ \text{m/s}$ .
Converting the mass gives $kg160\ \text{g} = 0.160\ \text{kg}$ .
Squaring the speed gives $v^{2} = 18^{2} = 324$ .
Substituting gives $\times 0.160 \times 324$ .
The kinetic energy store holds $25.9\ \text{J}$ to three significant figures.

Example

Speed of a falling stone

A stone is dropped from a bridge $m20\ \text{m}$ above a river, with $10\ \text{N/kg}$ and air resistance small enough to ignore.
The gravitational store emptied equals the kinetic store filled, so $12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h$ .
Cancelling the mass and rearranging gives $\sqrt{2 \times g \times \Delta h}$ .
Substituting gives $\sqrt{2 \times 10 \times 20} = \sqrt{400}$ .
The stone reaches the water at $20\ \text{m/s}$ .

Example

Finding a speed from the store

A trolley of mass $kg2.5\ \text{kg}$ holds $J45\ \text{J}$ in its kinetic energy store.
Rearranging gives $\sqrt{\dfrac{2 \times KE}{m}}$ .
Substituting gives $\sqrt{\dfrac{2 \times 45}{2.5}} = \sqrt{36}$ .
The trolley is moving at $6.0\ \text{m/s}$ .

Exam technique

Working with the squared speed

Write $\dfrac{1}{2} \times m \times v^{2}$ and square the speed on its own line so the working can be followed.
Square only the speed, never the mass and never the one half.
Keep the full value in the calculator when a rearranged answer is wanted, then take the square root last.
Compare two kinetic energy stores by comparing the masses and the squares of the speeds rather than the speeds themselves.
Round to the same number of significant figures as the data in the question, which is usually two or three.

Common Mistake

Do not multiply by the speed once instead of squaring it, because that badly underestimates the store.
Do not leave the mass in grams, because using grams in place of kilograms makes the answer a thousand times too large.
Do not forget the factor of $12\dfrac{1}{2}$ , which is the most commonly dropped mark in this calculation.
Do not say that a heavier object always holds more kinetic energy, because a light object moving quickly can hold more.

Self review

State what kinetic energy is and give its unit.
Write the equation for kinetic energy and name each quantity in it.
Explain what happens to the kinetic energy store when the speed triples.
Calculate the kinetic energy of a $kg1200\ \text{kg}$ car travelling at $m/s15\ \text{m/s}$ .
Explain why the landing speed of a dropped object does not depend on its mass.

The kinetic energy store

Definition

Kinetic energy

Kinetic energy is energy stored by an object because it is moving.

Kinetic energy is stored by an object because it is moving, so every moving object holds some.
The size of the store depends on the object's mass and on its speed.
Speed matters more than mass, because the store depends on the square of the speed.
A stationary object has a speed of $m/s0\ \text{m/s}$ , so its kinetic energy store holds $J0\ \text{J}$ .
The store fills as an object speeds up and empties as it slows down.
Kinetic energy is measured in joules, $J\text{J}$ , like every other store.

The kinetic energy equation

The store is calculated from $\dfrac{1}{2} \times m \times v^{2}$ .
Here $K E$ is the kinetic energy in joules, $m$ is the mass in kilograms and $v$ is the speed in metres per second, $m/s\text{m/s}$ .
Square the speed before multiplying, so a speed of $m/s6.0\ \text{m/s}$ gives $m2/s2v^{2} = 36\ \text{m}^{2}/\text{s}^{2}$ .
Writing the working as $\times m \times v^{2}$ makes the factor of one half harder to forget.
Rearranging for speed gives $\sqrt{\dfrac{2 \times KE}{m}}$ , and rearranging for mass gives $\dfrac{2 \times KE}{v^{2}}$ .
Doubling the mass at the same speed doubles the store, because the mass is not squared.
Doubling the speed at the same mass makes the store four times larger, since $2^{2} = 4$ .
Tripling the speed makes the store nine times larger, since $3^{2} = 9$ .
Convert grams into kilograms, and convert any speed given in $km/h\text{km/h}$ into $m/s\text{m/s}$ , before substituting.

Kinetic energy in falling and braking

A falling object empties its gravitational potential energy store and fills its kinetic energy store, so $12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h$ when resistive forces are small.
Cancelling the mass gives $\sqrt{2 \times g \times \Delta h}$ , which shows that the landing speed does not depend on how heavy the object is.
A vehicle's kinetic energy store has to be emptied before it stops, and the work done by the braking force equals that store, so $\times d = \dfrac{1}{2} \times m \times v^{2}$ .
Because the speed is squared, doubling a car's speed makes its kinetic energy store four times larger and so needs about four times the distance to stop with the same braking force.
Those joules are transferred into the thermal energy stores of the brakes, tyres and road, which is why brake discs become hot.

Example

Kinetic energy of a cyclist

A cyclist and bicycle have a combined mass of $kg80\ \text{kg}$ and travel at $m/s6.0\ \text{m/s}$ .
The equation is $\dfrac{1}{2} \times m \times v^{2}$ .
Squaring the speed gives $v^{2} = 6.0^{2} = 36$ .
Substituting gives $\times 80 \times 36$ .
The kinetic energy store holds $1440\ \text{J}$ .

Example

A mass given in grams

A cricket ball of mass $g160\ \text{g}$ is thrown at $m/s18\ \text{m/s}$ .
Converting the mass gives $kg160\ \text{g} = 0.160\ \text{kg}$ .
Squaring the speed gives $v^{2} = 18^{2} = 324$ .
Substituting gives $\times 0.160 \times 324$ .
The kinetic energy store holds $25.9\ \text{J}$ to three significant figures.

Example

Speed of a falling stone

A stone is dropped from a bridge $m20\ \text{m}$ above a river, with $10\ \text{N/kg}$ and air resistance small enough to ignore.
The gravitational store emptied equals the kinetic store filled, so $12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h$ .
Cancelling the mass and rearranging gives $\sqrt{2 \times g \times \Delta h}$ .
Substituting gives $\sqrt{2 \times 10 \times 20} = \sqrt{400}$ .
The stone reaches the water at $20\ \text{m/s}$ .

Example

Finding a speed from the store

A trolley of mass $kg2.5\ \text{kg}$ holds $J45\ \text{J}$ in its kinetic energy store.
Rearranging gives $\sqrt{\dfrac{2 \times KE}{m}}$ .
Substituting gives $\sqrt{\dfrac{2 \times 45}{2.5}} = \sqrt{36}$ .
The trolley is moving at $6.0\ \text{m/s}$ .

Exam technique

Working with the squared speed

Write $\dfrac{1}{2} \times m \times v^{2}$ and square the speed on its own line so the working can be followed.
Square only the speed, never the mass and never the one half.
Keep the full value in the calculator when a rearranged answer is wanted, then take the square root last.
Compare two kinetic energy stores by comparing the masses and the squares of the speeds rather than the speeds themselves.
Round to the same number of significant figures as the data in the question, which is usually two or three.

Common Mistake

Do not multiply by the speed once instead of squaring it, because that badly underestimates the store.
Do not leave the mass in grams, because using grams in place of kilograms makes the answer a thousand times too large.
Do not forget the factor of $12\dfrac{1}{2}$ , which is the most commonly dropped mark in this calculation.
Do not say that a heavier object always holds more kinetic energy, because a light object moving quickly can hold more.

Self review

State what kinetic energy is and give its unit.
Write the equation for kinetic energy and name each quantity in it.
Explain what happens to the kinetic energy store when the speed triples.
Calculate the kinetic energy of a $kg1200\ \text{kg}$ car travelling at $m/s15\ \text{m/s}$ .
Explain why the landing speed of a dropped object does not depend on its mass.

8.2.2 Kinetic energy

The kinetic energy store

The kinetic energy equation

Kinetic energy in falling and braking

Kinetic energy of a cyclist

A mass given in grams

Speed of a falling stone

Finding a speed from the store

Working with the squared speed

Review 8.2.2 Kinetic energy by teaching Genie

8.2.2 Kinetic energy

The kinetic energy store

The kinetic energy equation

Kinetic energy in falling and braking

Kinetic energy of a cyclist

A mass given in grams

Speed of a falling stone

Finding a speed from the store

Working with the squared speed

Review 8.2.2 Kinetic energy by teaching Genie

8.2.2 Kinetic energy

The kinetic energy store

The kinetic energy equation

Kinetic energy in falling and braking

Kinetic energy of a cyclist

A mass given in grams

Speed of a falling stone

Finding a speed from the store

Working with the squared speed

Review 8.2.2 Kinetic energy by teaching Genie

8.2.2 Kinetic energy Revision Guide

8.2.2 Kinetic energy

The kinetic energy store

The kinetic energy equation

Kinetic energy in falling and braking

Kinetic energy of a cyclist

A mass given in grams

Speed of a falling stone

Finding a speed from the store

Working with the squared speed

Review 8.2.2 Kinetic energy by teaching Genie

8.2.2 Kinetic energy Revision Guide