A.2.2 Linear momentum and impulse Notes

Understanding Momentum and Impulse

1. You're playing a game of billiards.
2. You strike the cue ball, and it collides with another ball, sending both rolling across the table.
3. What determines how these balls move after the collision?

The answer lies in two fundamental concepts: momentum and impulse.

What is Momentum?

Definition of Momentum

Momentum is defined as the product of an object's mass and its velocity:

$$
\vec{p} = m \vec{v}
$$

where $\vec{p}$ is the momentum (in $\text{kg m s}^{-1}$), $m$ is the mass (in $\text{kg}$), $\vec{v}$ is the velocity (in $\text{m s}^{-1}$).

Newton originally expressed his Second Law in terms of momentum rather than acceleration.

He stated that:

"the rate of change of momentum of an object is proportional to the force applied, and takes place in the direction of the force."

This can be written as:

$$
\vec{F} = \frac{\Delta\vec{p}}{\Delta t}
$$

1. Continuing from example about the rocket, since its mass is decreasing, the simple form $$\vec{F} = m \vec{a}$$ is not sufficient.
2. Instead, we rely on: $$\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$$
3. It could be written more rigorously as: $$\vec{F} = \frac{d}{dt}(m \vec{v})$$ which accounts for both the changing velocity and changing mass of the rocket.
4. When the mass is constant, this equation simplifies to the more familiar form: $$\vec{F} = m \frac{d\vec{v}}{dt} = m \vec{a}$$

Why is Momentum Important?

1. Momentum helps us understand how objects behave during interactions like collisions or explosions.
2. It is a conserved quantity (we prove it further), meaning the total momentum of a system remains constant if no external forces act on it.

Impulse: Changing Momentum

Impulse-Momentum Theorem

1. The impulse–momentum theorem relates the force acting on an object to the change in its momentum: $\vec{F} \Delta t = \Delta \vec{p}$$
2. Impulse is defined as the product of force and the time interval over which the force acts: $$\vec{J} = \vec{F} \Delta t$$

1. If the force is not constant, we shall sum product of force and small time interval (small enough to regard the force acting on a body during it to be constant) over the time interval during which the force is applied to the body. $$\vec{J} = \Delta \vec{p} = \sum \vec{F} \Delta t$$
2. Mathematically, in the limit $\Delta t \to 0$ the impulse is the integral of force over time: $$\vec{J} = \int \vec{F} \, dt$$

Calculating Impulse

Impulse can be calculated in two ways:

1. Using the formula $J = F \Delta t$ for constant forces.
2. By finding the area under a force-time graph for variable forces.

Conservation of Momentum

1. We shall thus prove the principle of conservation of momentum.
2. In an isolated system, where no external forces act, the total momentum of the system remains constant.
3. To see why this is true, we start with Newton’s Second Law in its original form: $$\vec{F}_{\text{net}} = \frac{\Delta \vec{p}}{\Delta t}$$
4. If there are no external forces acting on the system, then: $$\vec{F}_{\text{net}} = 0$$
5. This implies: $$\frac{\Delta\vec{p}}{\Delta t} = 0$$
6. This means that the rate of change of momentum is zero.
7. Therefore, the total momentum of the system does not change with time: $$\vec{p}_{\text{total}} = \text{constant}$$
8. This means that in any interaction (such as a collision or explosion) within a isolated system (collection of objects that do not interact with anything outside of that collection), the total momentum before the interaction equals the total momentum after the interaction: $$\sum \vec{p}_{\text{before}} = \sum \vec{p}_{\text{after}}$$