Kinetic Energy

Derivation of KE = 1/2mv2

Kinetic energy is energy an object has due to its motion (or velocity)
A force can make an object accelerate; work is done by the force and energy is transferred to the object
Using this concept of work done and an equation of motion, the extra work done due to an object's speed can be derived

Consider a mass, m, at rest which accelerates to a speed, v, over a distance, d
The work done in accelerating the mass is:

$W = F d$

Newton's Second Law states that:

$F = m a$

The suvat equation for velocity is:

$v^{2} = u^{2} + 2 a s$

- If the initial speed is zero, u = 0
- And distance s = d
- Then:

$v^{2} = 2 a d$

- Rearranging to make a the subject:

$a = \frac{v^{2}}{2 d}$

Substituting this expression for a into Newton's Second Law gives:

$F = m a = \frac{m v^{2}}{2 d}$

Substituting this expression for F into the work done equation gives:

$W = \frac{m v^{2}}{2 d} \times d = \frac{1}{2} m v^{2}$

The mass is now able to do extra work due to its speed
- The amount of extra work is equal to $\frac{1}{2} m v^{2}$
The mass now has:

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

Kinetic energy is energy an object has due to its motion (or velocity)
- The faster an object is moving, the greater its kinetic energy
When an object is falling, it is gaining kinetic energy since it is gaining speed. This energy transferred from the gravitational potential energy it is losing
An object will maintain this kinetic energy unless its speed changes

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

Where:
- E_k = kinetic energy in joules (J)
- m = mass in kilograms (kg)
- v = velocity in metres per second (m s^-1)

A Car Travelling Forwards

Kinetic energy diagram, downloadable AS & A Level Physics revision notes

Kinetic energy is the energy an object has when its moving determined by its mass and its speed

Worked example

A body travelling with a speed of 12 m s^-1 has kinetic energy 1650 J. The speed of the body is increased to 45 m s^-1. Determine the body's new kinetic energy.

Answer:

Step 1: List the known quantities

Initial speed, v_i = 12 m s^-1
Initial kinetic energy, E_{k i} = 1650 J
Final speed, v_f = 45 m s^-1

Step 2: State the kinetic energy equation

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

Step 3: Determine the object's mass

The mass will not change
Therefore, mass can be calculated from the initial kinetic energy

$m = \frac{2 E_{k i}}{{v_{i}}^{2}}$

$m = \frac{2 \times 1650}{12^{2}}$

$m = 23 kg$

Step 4: Substitute the known values to calculate the final kinetic energy

$E_{k f} = \frac{1}{2} m {v_{f}}^{2}$

$E_{k f} = 0.5 \times 23 \times 45^{2}$

$E_{k f} = 23 000 J$

Exam Tip

If a question asks about the ‘loss of kinetic energy’, don't be tempted to include a negative sign since energy is a scalar quantity.

Kinetic Energy (CIE A Level Physics)

Revision Note

Derivation of KE = 1/2mv2

Kinetic Energy

A Car Travelling Forwards

Worked example

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Author: Leander