Understanding Momentum and Impulse
- You're playing a game of billiards.
- You strike the cue ball, and it collides with another ball, sending both rolling across the table.
- What determines how these balls move after the collision?
The answer lies in two fundamental concepts: momentum and impulse.
What is Momentum?
Definition
Momentum
Momentum is a measure of how difficult it is to stop a moving object. It depends on two factors: the object's mass and its velocity.
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:
$$
\vec{F} = \frac{\Delta\vec{p}}{\Delta t}
$$
- This form is particularly useful when dealing with systems where the mass is not constant, such as a rocket.
- As a rocket burns fuel, it ejects mass backward and becomes lighter over time.
- Even though the total system’s mass is changing, the force generated by the rocket engines can still be understood as the rate of change of momentum.
- For example, if a rocket expels gas backward at high velocity, the momentum of the expelled gas increases in one direction, and by conservation of momentum, the rocket gains momentum in the opposite direction.
- This causes the rocket to accelerate.
- Continuing from example about the rocket, since its mass is decreasing, the simple form $$\vec{F} = m \vec{a}$$ is not sufficient.
- Instead, we rely on: $$\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$$
- 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.
- 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?
- Momentum helps us understand how objects behave during interactions like collisions or explosions.
- 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.
- Momentum is a vector quantity, so direction must be considered when applying this principle.
- When convenient, one may define the axes in the framework of the problem and work with projections instead.
Impulse: Changing Momentum
Definition
Impulse
Impulse describes how a force applied over a period of time changes an object's momentum.
