Kinetic Energy (Cambridge International A-Level Physics)

Derivation of KE = 1/2mv2

Derivation of KE = 1/2mv²

• Kinetic energy is energy an object has due to its motion (or velocity)

• Resultant forces make objects accelerate as described by Newton's second law F = ma

• Work is done by that 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 velocity, v, over a displacement, s

• The work done in accelerating the mass is:

$W=Fs$

• Newton's Second Law states that:

$F=ma$

• The suvat equation for velocity is: $v^{2}=u^{2}+2as$

  • If the initial velocity is zero, u = 0
  • And displacement, s
  • Then: $v^{2}=2as$
  • Rearranging to make a the subject:

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

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

$F=ma=\frac{mv^{2}}{2s}$

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

$W=\frac{mv^{2}}{2s}\times s=\frac{1}{2}mv^{2}$

• The mass is now able to do extra work due to its speed

  • The amount of extra work is equal to $\frac{1}{2}mv^{2}$

• The mass now has:

$E_{k}=\frac{1}{2}mv^{2}$

Kinetic Energy

• 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

• The amount of kinetic energy an object has is determined by its mass and its speed:

$E_{k}=\frac{1}{2}mv^{2}$

• 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 is the energy an object has when it is 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

$E_{k}=\frac{1}{2}mv^{2}$

Step 3: Determine the object's mass

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

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

$m=\frac{2\times1650}{12^{2}}$

$m=23\mathrm{kg}$

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

$E_{kf}=\frac{1}{2}mv_{f}^{2}$

$E_{kf}=0.5\times23\times45^{2}$

$E_{kf}=23000J$