# Calculation of energy changes

## Calculating kinetic energy

The amount of in a moving object can be calculated using the equation:

$kinetic~energy= \frac{1}{2}\times mass \times speed^2$

$E_{k} = \frac{1}{2}~m~v^{2}$

This is when:

• kinetic energy (Ek) is measured in joules (J)
• mass (m) is measured in kilograms (kg)
• speed (v) is measured in metres per second (m/s)

### Example

An apple of mass 100 g falls from a tree. It reaches a speed of 6 m/s before landing on Isaac’s head. What is the gain of kinetic energy of the apple?

$E_{k} = \frac{1}{2}~m~v^{2}$

$E_{k} = \frac{1}{2} \times 0.1 \times6^{2}$

$E_{k} = \frac{1}{2} \times 0.1 \times 36$

$E_{k} = 1.8~J$

Question

How much kinetic energy does a 30 kg dog have when it runs at 4 m/s?

$E_{k} = \frac{1}{2}~m~v^{2}$

$E_{k} = \frac{1}{2} \times 30 \times 4^{2}$

$E_{k} = \frac{1}{2} \times 30 \times 16$

$E_{k} = 240~J$

## Calculating elastic potential energy

The amount of stored in a stretched spring can be calculated using the equation:

$elastic~potential~energy= \frac{1}{2} \times spring~constant \times extension^{2}$

$E_{e}=\frac{1}{2}~k~e^{2}$

This is when:

• elastic potential energy (Ee) is measured in joules (J)
• spring constant (k) is measured in newtons per metre (N/m)
• extension (e) is measured in metres (m)

### Example

Robert stretches a spring with a spring constant of 3 N/m until it is extended by 50 cm. What is the elastic potential energy stored by the spring?

$E_{e} = \frac{1}{2}~k~e^{2}$

$E_{e} = \frac{1}{2} \times 3\times 0.5^{2}$

$E_{e} = \frac{1}{2} \times 3\times 0.25$

$E_{e} = 0.375~J$

Question

How much elastic potential energy does a spring store when it is compressed by 0.2 m if it has a spring constant of 5 N/m?

$E_{p} = \frac{1}{2}~k~e^{2}$

$E_{e} = \frac{1}{2} \times 5 \times 0.2^{2}$

$E_{e} = \frac{1}{2} \times 5 \times 0.04$

$E_{e} = 0.1~J$

## Calculating gravitational potential energy

The amount of stored by an object at height can be calculated using the equation:

Gravitational potential energy = mass × gravitational field strength × height

$E_{p}=m~g~h$

This is when:

• gravitational potential energy (Ep) is measured in joules (J)
• mass (m) is measured in kilograms (kg)
• gravitational field strength (g) is measured in newtons per kilogram (N/kg
• height (h) is measured in metres (m)

### Example

Galileo takes a 5 kg cannonball to the top of the Tower of Pisa for one of his experiments. The tower is 56 m high. How much gravitational potential energy has the cannonball gained? (g = 10 N/kg)

$E_{p}=m~g~h$

$E_{p}= 5 \times 10 \times 56$

$E_{p}= 2,800 J$

Question

How much gravitational potential energy does a 500 g book gain when it is lifted up 1.5 m onto a shelf?

$E_{p}= m~g~h$

$E_{p}= 0.5 \times 10 \times 1.5$

$E_{p}= 7.5~J$

For any of these equations you may need to change the subject of the formula.

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