4.1.6 Momentum

Momentum is the product of an object's mass and its velocity. It is given by the formula:

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

Momentum is always conserved in any interaction where no external forces are present. This principle is known as the Conservation of Momentum, which states that the total momentum before an event (such as a collision) is equal to the total momentum after the event.

Worked Example – Conservation of Momentum

A car of mass 500 kg, moving at 4 m/s, collides with a stationary truck of mass 1500 kg. They join together and move with a combined velocity $V$ . Find $V$.

1. Calculate the total momentum before the collision:

$$\text{Momentum before} = (500 \times 4) + (1500 \times 0) = :highlight[2000] \, \text{kg m/s}$$

2. Apply conservation of momentum:

$$\text{Total momentum before} = \text{Total momentum after}$$ $$2000 = (500 + 1500) \times V$$ $$V = :success[1 \, \text{m/s}]$$

Thus, the combined velocity $V$ after the collision is 1 m/s.

Impulse

Newton's 2nd law can be rephrased to show that force is the rate of change of momentum. Given by:

$$F = \frac{\Delta (mv)}{\Delta t}$$

Rearranging, we get:

$$F \Delta t = \Delta (mv)$$

Where:

• $F \Delta t$is known as impulse.
• Impulse represents the change in momentum over a specific period of time. The area under a force-time graph is also equal to the change in momentum.

Applications of Impulse in Safety

Impulse calculations are essential in designing safety features like crumple zones, seat belts, and airbags in cars. These features increase the impact time, which reduces the force experienced by passengers, decreasing the risk of injury.

Types of Collisions

1. Elastic Collisions

• Both momentum and kinetic energy are conserved.

2. Inelastic Collisions

• Only momentum is conserved. Some kinetic energy is transformed into other forms (e.g., heat, sound).
• If objects stick together after collision, it is classified as an inelastic collision.