Force & Momentum

Momentum Basics

Momentum is a measure of how difficult it is to stop a moving object. It depends on both the mass of the object and its velocity.

Definition: Momentum is the product of an object's mass and its velocity.

Mathematically, momentum $p$ is given by:

$$p = m \times v$$

where:

• $p$ = momentum (kg·m/s)
• $m$ = mass (kg)
• $v$ = velocity (m/s)

Momentum is a vector quantity, which means it has both magnitude and direction. The direction of momentum is the same as the direction of velocity.

The unit of momentum is kilogram metre per second (kg·m/s).

For example, a 2 kg ball moving at 3 m/s has momentum 6 kg·m/s.

For instance, if a car of mass 1000 kg is moving at 20 m/s east, its momentum is:

\[ p = 1000 \times 20 = 20,000 \text{ kg·m/s east} \]

Force and Momentum Relationship

A force acting on an object changes its momentum. The greater the force or the longer it acts, the bigger the change in momentum.

Impulse is defined as the product of force and the time for which it acts:

$$\text{Impulse} = F \times t$$

where:

• $F$ = force (newtons, N)
• $t$ = time (seconds, s)

Impulse is equal to the change in momentum of the object and is also a vector quantity, having the same direction as the change in momentum:

$$F \times t = \Delta p = m v_{\text{final}} - m v_{\text{initial}}$$

This relationship explains why safety features like airbags and crumple zones in cars are effective: they increase the time over which the force acts, reducing the force and therefore the risk of injury.

For example, if a force of 500 N acts on a stationary object for 4 seconds, the impulse is:

\[ \text{Impulse} = 500 \times 4 = 2000 \text{ N·s} \]

This means the object's momentum changes by 2000 kg·m/s.

Conservation of Momentum

In a closed system (where no external forces act), the total momentum before an event is equal to the total momentum after the event. This is known as the conservation of momentum.

Mathematically:

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

This principle applies especially in collisions and explosions.

Types of collisions:

• Elastic collisions: Both momentum and kinetic energy are conserved. 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, and objects may stick together.

For example, when two ice hockey pucks collide on a frictionless surface, their total momentum before and after the collision remains the same.

Momentum in Collisions

To calculate momentum before and after a collision, multiply each object's mass by its velocity, taking direction into account.

Using the conservation of momentum, you can solve problems where velocities or masses change after collisions.

For example, if two trolleys collide and stick together (an inelastic collision), their combined momentum after the collision equals the sum of their individual momenta before the collision.

Safety applications of momentum include:

• Seat belts: Stretch slightly to increase the time over which the force acts, reducing the force on the passenger.
• Airbags: Inflate to cushion the passenger and increase the impact time.
• Crumple zones: Designed to deform in a collision, increasing the time taken to stop the car and reducing force.

These features reduce the force experienced by passengers by increasing the time over which momentum changes.

For instance, if a 1500 kg car travelling at 20 m/s crashes and comes to rest in 0.5 seconds, the average force exerted on the car can be calculated using impulse:

$$F = \frac{\Delta p}{t} = \frac{1500 \times (0 - 20)}{0.5} = \frac{-30,000}{0.5} = -60,000 \text{ N}$$

The negative sign shows the force acts opposite to the car's initial direction.