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Elastic & Inelastic Collisions (CIE AS Physics)

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Leander

Last updated

15 October 2024

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:

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

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

Kinetic energy depends on the speed of an object
In a perfectly elastic collision (such as a head-on 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 collision. What is the total kinetic energy after the impact?

WE - Elastic collision question image, downloadable AS & A Level Physics revision notes

Answer:

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

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

$E_{k b e f o r e} = E_{k a f t e r}$

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

$E_{k b e f o r e} = \frac{1}{2} m v^{2} + \frac{1}{2} m v^{2}$

$E_{k b e f o r e} = m v^{2} = E_{k a f t e r}$

Therefore, the correct answer is C

Examiner Tip

Despite velocity being a vector, kinetic energy is a scalar quantity and therefore will never include a minus sign.

This is because in the kinetic energy formula, mass is scalar and the v² will always give a positive value whether its a negative or positive velocity.

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
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

$E_{k b e f o r e} = \frac{1}{2} m_{x} {v_{x}}^{2} + 0$

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

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

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

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

Step 3: Compare the expressions and determine if they are equal

$\frac{1}{2} m_{x} {v_{x}}^{2} \neq \frac{1}{2} (2 m) {v_{x + y}}^{2}$

$E_{k b e f o r e} \neq E_{k a f t e r}$

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

Examiner Tip

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

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Elastic & Inelastic Collisions (CIE AS Physics)

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Author: Leander