Conservation of Momentum

Principle of Conservation of Momentum

Momentum is a vector quantity, meaning it has both magnitude and direction. The principle of conservation of momentum states that in a closed system (where no external forces act), the total momentum before an event is equal to the total momentum after the event.

Mathematically, this is expressed as:

$$\sum \vec{p}_{\text{before}} = \sum \vec{p}_{\text{after}}$$

A closed system means that the objects involved do not gain or lose momentum to anything outside the system. This principle applies to all interactions, including collisions and explosions.

Since momentum is a vector, direction matters. For example, if two objects move towards each other, their momenta have opposite directions and must be considered accordingly.

For instance, if a 2 kg ball moves right at 3 m/s and a 3 kg ball moves left at 2 m/s, the total momentum before interaction is:

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

The total momentum is zero because the momenta cancel out. After interaction, the total momentum must still be zero.

Collisions and Momentum

Collisions are common examples where conservation of momentum is applied. There are two main types:

• Elastic collisions: Both momentum and kinetic energy are conserved. The objects bounce off each other without permanent deformation or heat generation.
• Inelastic collisions: Momentum is conserved, but kinetic energy is not. Some energy is converted into other forms, like heat or sound, or causes deformation.

Examples:

• Billiard balls colliding are nearly elastic collisions, as they bounce off with little energy loss.
• Car crashes are usually inelastic; cars crumple and kinetic energy converts to sound, heat, and deformation.

In all collisions, the total momentum before and after remains the same, regardless of energy changes.

For example, two cars collide and stick together (a perfectly inelastic collision). The combined mass moves with a velocity that conserves the total momentum.

Calculating Momentum Changes

Momentum (p) is calculated by:

$p = m \times v$

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

The change in momentum of an object is related to the force applied and the time for which it acts:

$\text{Change in momentum} = \text{Force} \times \text{Time}$

This is also called impulse. Impulse has the same units as momentum (kg m/s) and represents the effect of a force acting over a period of time.

For example, if a force of 10 N acts on a ball for 0.5 seconds, the change in momentum is:

$\Delta p = 10 \times 0.5 = 5 \text{ kg m/s}$

This means the ball’s momentum increases by 5 kg m/s in the direction of the force.

Impulse is useful for understanding how forces affect motion, especially in collisions or when objects start or stop moving.

For example, car safety features like airbags increase the time over which the force acts during a crash, reducing the force and therefore the risk of injury.

Example: A 0.15 kg tennis ball moving at 20 m/s is caught and brought to rest in 0.05 seconds. Calculate the average force exerted on the ball.

First, calculate the change in momentum:

$\Delta p = m \times (v_{\text{final}} - v_{\text{initial}}) = 0.15 \times (0 - 20) = -3 \text{ kg m/s}$

The negative sign shows the momentum decreases.

Using impulse formula:

$F = \frac{\Delta p}{t} = \frac{-3}{0.05} = -60 \text{ N}$

The force is 60 N opposite to the ball’s initial motion.