Calculation of energy changes

Calculating kinetic energy

The amount of kinetic energy 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 elastic potential energy 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 gravitational potential energy 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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