Collisions

Types of Collisions

Collisions describe events where two or more objects come into contact and exert forces on each other for a short time. There are two main types of collisions:

• Elastic collisions: Both momentum and kinetic energy are conserved. The objects bounce off each other without any permanent deformation or generation of heat. An example is two snooker balls hitting each other.
• Inelastic collisions: Momentum is conserved, but kinetic energy is not. Some kinetic energy is converted into other forms such as heat, sound, or deformation. An example is a car crash where the vehicles crumple.

Everyday examples include:

• Elastic: Billiard balls colliding, a tennis ball bouncing off the ground.
• Inelastic: Clay blobs sticking together after impact, car crashes.

Conservation of Momentum

Momentum is a vector quantity defined as the product of an object's mass and velocity:

$\text{momentum} = \text{mass} \times \text{velocity}$

The law of conservation of momentum states that in a closed system (where no external forces act), the total momentum before a collision equals the total momentum after the collision:

$$\text{Total momentum before} = \text{Total momentum after}$$

Because momentum is a vector, direction matters. When calculating total momentum, you must consider the direction of velocity (positive or negative). For example, if two objects move towards each other, one velocity may be positive and the other negative.

A closed system means no external forces like friction or air resistance affect the objects during the collision. This allows momentum to be conserved.

Calculating Momentum in Collisions

Momentum is calculated using the formula:

$$\text{momentum} (p) = m \times v$$

where $m$ is mass in kilograms (kg) and $v$ is velocity in metres per second (m/s).

In collisions, you can use conservation of momentum to find unknown velocities or masses:

$$\sum p_{\text{before}} = \sum p_{\text{after}}$$

For collisions in one dimension (straight line), assign one direction as positive and the opposite as negative. Then write the momentum before and after collision and solve for the unknown.

For instance, if a 2 kg ball moving at 3 m/s collides head-on with a 3 kg ball moving at -2 m/s (opposite direction), the total momentum before collision is:

$$(2 \times 3) + (3 \times -2) = 6 - 6 = 0 \text{ kg m/s}$$

After the collision, the total momentum must still be 0 kg m/s.

Example: If after collision the 2 kg ball moves at 1 m/s, what is the velocity of the 3 kg ball?

Using conservation of momentum:

$$(2 \times 1) + (3 \times v) = 0$$

$$2 + 3v = 0$$

$$3v = -2$$

$$v = -\frac{2}{3} = -0.67 \text{ m/s}$$

The negative sign shows it moves in the opposite direction to the 2 kg ball.

Momentum and Safety

When momentum changes (such as during a collision), a force is exerted on the object. The force depends on how quickly the momentum changes:

$$\text{Force} = \frac{\text{Change in momentum}}{\text{Time taken}}$$

If the time over which the collision happens is increased, the force experienced is reduced. This is why safety features in vehicles are designed to increase collision time and reduce forces on passengers.

Examples include:

• Seat belts: Stretch slightly during a crash, increasing the time over which the wearer’s momentum changes, reducing the force on the chest.
• Airbags: Inflate to cushion the impact, increasing collision time and spreading the force over a larger area.

By increasing the time taken for the momentum to change, these safety devices reduce the risk of injury by lowering the force exerted on the body.