Lesson 2 of 5Energy and Society

Unit 3 · Lesson 2 of 5

Kinetic Energy, Potential Energy and Conservation

The energy of motion and the energy of position, and the law that lets you solve problems kinematics cannot touch.

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Key things to remember

1 $K E = \frac{1}{2} m v^{2}$ — doubling the speed quadruples the kinetic energy.
2Kinetic energy is a scalar and is never negative, because the velocity is squared.
3 $PE = m g h$ , where $h$ is the vertical height above a reference level you choose and state.
4Energy is never created or destroyed, only transformed or transferred.
5Without friction, $K E_{i} + P E_{i} = K E_{f} + P E_{f}$ — mechanical energy is conserved.
6With friction, the missing mechanical energy has become thermal energy: $K E_{i} + P E_{i} = K E_{f} + P E_{f} + E_{t h er ma l} .$
7Energy methods ignore the path, which is what makes them work where kinematics cannot.

Energy is the capacity to do work, measured in joules. It comes in many forms, but two of them account for most of this unit: the energy something has because it is moving, and the energy it has because of where it is.

Kinetic energy

K E = \frac{1}{2} m v^{2}

Mass in kilograms, speed in metres per second, result in joules.

The squared velocity is the important part. Doubling the mass doubles the kinetic energy, but doubling the speed quadruples it. This single fact explains most of road safety: a car at $100 km/h$ carries four times the kinetic energy it had at $50 km/h$ , and brakes have to remove all of it.

Kinetic energy is a scalar and is never negative

The velocity is squared, so a negative velocity gives a positive energy. An object moving left at

10 m/s

has exactly the same kinetic energy as one moving right at

10 m/s

. Direction is simply not part of the quantity.

Gravitational potential energy

PE = m g h

Height measured from whatever reference level you choose — and say which.

Lift an object and you do work against gravity; that work is stored as gravitational potential energy, ready to be released if the object falls. Unlike kinetic energy, potential energy has no natural zero — you have to choose one. The floor, the ground, or the lowest point of the motion are all reasonable choices, and the change in $PE$ comes out the same whichever you pick.

h is the vertical height, not the distance travelled

A box dragged

10 m

up a ramp that rises

3 m

gains

m g h

with

h = 3 m

, not

10 m

. Gravity acts vertically, so only vertical displacement counts. This is also why walking around a level room changes your potential energy not at all.

The law of conservation of energy

Conservation of energy

Energy cannot be created or destroyed, only transformed from one form into another or transferred from one object to another. The total energy of an isolated system is constant.

Fig 1Energy changes form at every arrow. The total never changes.

In the absence of friction, the sum of kinetic and potential energy — the total mechanical energy — stays constant. A ball dropped from a height trades $PE$ for $K E$ joule for joule, so at every instant the two add to the same number.

E_{t o t a l} = K E + PE = constant ⟹ K E_{i} + P E_{i} = K E_{f} + P E_{f}

The form worth writing down, because it turns most problems into one line of algebra.

Fig 2The bars swap height as the cart descends, but their total is unchanged.

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Drop a cart from a height and watch kinetic, potential and thermal energy trade places while the total holds steady.

Why this beats kinematics

Conservation of energy does not care about the path. A ball dropped straight down, one rolled down a curved ramp and one on a loop-the-loop all arrive at the same height with the same speed, provided friction is negligible. Kinematics would need the acceleration at every point; energy needs only the start and the end.

When friction is present, mechanical energy is not conserved — but total energy still is. The missing mechanical energy has become thermal energy in the surfaces. Written as an equation: $K E_{i} + P E_{i} = K E_{f} + P E_{f} + E_{t h er ma l} .$

AMechanical energy conserved

No friction or air resistance
$K E_{i} + P E_{i} = K E_{f} + P E_{f}$
Speed at a given height is the same going up and coming down
A pendulum returns to exactly its release height

BFriction present

Surfaces or air remove mechanical energy
$K E_{i} + P E_{i} = K E_{f} + P E_{f} + E_{t h er ma l}$
Arrives slower than the frictionless prediction
A pendulum swings lower each time and eventually stops

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

1200 kg

car travels at

25 m/s

. Find its kinetic energy, and its kinetic energy at

50 m/s

Example 2

0.45 kg

ball is dropped from rest at a height of

12 m

. Using energy, find its speed just before it lands. Ignore air resistance.

Example 3

2.0 kg

cart is released from rest at the top of a

1.8 m

ramp and reaches the bottom at

5.0 m/s

. How much energy was lost to friction?

Example 4

A pendulum bob of mass

0.30 kg

is pulled aside until it is

0.25 m

above its lowest point and released. Find its maximum speed.

Where marks get lost

Forgetting to square the velocity, or squaring after multiplying by the mass. Square $v$ first.
Thinking doubling the speed doubles the kinetic energy. It quadruples it.
Using the distance along a ramp as $h$ in $PE = m g h$ . Only the vertical rise counts.
Never stating the reference level for potential energy, so the numbers cannot be checked.
Applying $K E_{i} + P E_{i} = K E_{f} + P E_{f}$ where friction is significant. Add the thermal term.
Claiming energy is 'lost' in a way that breaks conservation. It becomes thermal energy — still there, just no longer useful.
Giving kinetic energy a direction or a negative value. It is a scalar and always positive.

Flashcards 9

Kinetic energy

Effect of doubling speed on

K E

Gravitational potential energy

What is

h

PE = m g h

+ 5 more

Practice

What is the kinetic energy, in joules, of a

4.0 kg

object moving at

6.0 m/s

A car's speed doubles. What happens to its kinetic energy?

How much gravitational potential energy, in joules, does a

3.0 kg

book gain when lifted

1.5 m

An object is dropped from rest at

20.0 m

. Using energy conservation, what is its speed on landing, in m/s? Ignore air resistance.

m/s

A box is dragged

8.0 m

along a ramp that rises

2.0 m

vertically. Which value of

h

goes into

PE = m g h

A pendulum gradually swings lower and eventually stops. Which statement is correct?

0.50 kg

ball is thrown upward at

12 m/s

. Using energy, what maximum height in metres does it reach?

State the law of conservation of energy in one sentence.

Match each situation to the energy transformation taking place.

A ball falling from a shelf

A car braking to a stop

A cyclist pedalling up a hill

An LED bulb operating

A cart is released from rest at the top of a frictionless ramp. Order these moments by the cart's kinetic energy, largest first.

1One quarter of the way up from the bottom
2At the moment of release at the top
3Halfway up the ramp
4At the bottom of the ramp

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