Elastic & Inelastic Collisions (Cambridge International AS Physics)

Elastic Collisions

• When two objects collide, they may spring apart retaining all of the kinetic energy of the system

• This would be a perfect elastic collision

• In an elastic collision, all of the kinetic energy is conserved

• Recall the kinetic energy equation:

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

• Where:

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

• Kinetic energy depends on the speed of an object

• In a perfectly elastic collision:

the relative speed of approach = the relative speed of separation

Worked Example Two similar spheres, each of mass m and velocity v are travelling towards each other.

The spheres have a head-on elastic collision. What is the total kinetic energy after the impact?

Answer:

Step 1: Equate the kinetic energy before and after the collision

• In an elastic collision, the kinetic energy of the system is conserved

$E_{kbefore}=E_{kafter}$

Step 2: Write an expression for the kinetic energy before the collision

$E_{kbefore}=\frac{1}{2}mv^{2}+\frac{1}{2}mv^{2}$

$E_{kbefore}=mv^{2}=E_{kafter}$

• Therefore, the correct answer is C

Exam Hint Despite velocity being a vector, kinetic energy is never negative because v² is always positive.

Inelastic Collisions

• Whilst the momentum of a system is always conserved in interactions between objects, kinetic energy is not always conserved

• An inelastic collision is one where kinetic energy is not conserved

• The kinetic energy is transferred to other energy stores

• Inelastic collisions occur when two objects collide and they crumple and deform

• If the total momentum is zero, all of the kinetic energy of the system may be transferred away from the system and the objects will come to a halt

• Or some of the kinetic energy of the system may be transferred away and the objects will move as one body at a slower speed than the original objects

• A perfectly inelastic collision is when two objects stick together after collision

Worked Example Two trolleys X and Y are of equal mass. Trolley X moves towards trolley Y which is initially stationary.

After the collision, the trolleys join and move together.

Prove that this collision is inelastic.

Answer:

Step 1: Write an expression for the kinetic energy of the system before the collision

$E_{kbefore}=\frac{1}{2}m_{x}v_{x}^{2}+0$

• Object Y is stationary before the collision, so its kinetic energy is zero

Step 2: Write an expression for the kinetic energy of the system after the collision

$E_{k a f t e r} = 1 \frac{1}{2} ( m_{x + m_{y} ) v_{x} + y}^{2}$

• Both trolleys are of equal mass, therefore m_x + m_y = 2m

$E_{k a f t e r} = 1 \frac{1}{2} ( 2 m ) v_{x + y}^{2}$

Step 3: Use conservation of momentum to find the speed after the collision

$mv_{x}=2mv_{x+y}$

$v_{x+y}=\frac{v_{x}}{2}$

Step 4: Compare the expressions and determine if they are equal

$E_{k} after = \frac{1}{2} ( 2 m ) ( v_\frac{{x}}{2} ) ^{2} = \frac{1}{4} m v_{x}^{2}$

$E_{kbefore}=\frac{1}{2}mv_{x}^{2}$

$E_{kafter}=\frac{1}{2}E_{kbefore}$

$E_{kbefore}\neq E_{kafter}$

• The kinetic energy before the collision is not equal to the kinetic energy after the collision
• Therefore, the collision is inelastic

Exam Hint Although kinetic energy may not always be conserved, remember momentum will always be conserved.