Topic 4.2 Notes – Change in Momentum and Impulse

1. Momentum and Its Rate of Change

Linear momentum

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

• Vector quantity. Same direction as velocity.
• Units: $\text{kg}\cdot\text{m/s}$
• For a system: total momentum is the vector sum of each object’s momentum.

If two objects move in opposite directions, their momenta can partially cancel. Always think in components.

Net force and momentum

$$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}$$

This is the most general form of Newton’s second law.

What it means:

• A larger net force makes momentum change faster.
• If $\mathbf{F}_{\text{net}} = 0$, then momentum is constant. The object might still be moving at constant velocity.

Notice this works even if mass changes. That becomes important later.

Momentum-time graphs

Here’s what the relationship looks like graphically. In a momentum versus time graph, the slope at any point tells you the net force.

Momentum vs. time graph with varying net force

Key connection:

• Slope of a p-t graph = $F_{\text{net}}$
• Steeper slope → larger force
• Horizontal line → zero net force

If you ever see a momentum graph on a quiz, think “take the slope.”

2. Impulse

Impulse measures the total effect of a force over time.

Definition

$$\mathbf{J} = \int_{t_{1}}^{t_{2}} \mathbf{F}_{\text{net}}(t)\, dt$$

For constant force:

$$\mathbf{J} = \mathbf{F}_{\text{net}} \Delta t$$

• Units: $\text{N}\cdot\text{s} = \text{kg}\cdot\text{m/s}$
• Vector quantity, same direction as the net force

So impulse is the total “push” delivered.

You can get the same impulse from:

• Big force, short time
• Small force, long time

That’s why airbags work. Same change in momentum, longer time, smaller force.

Force-time graphs

Impulse shows up visually as area under a force-time graph.

Impulse as area under an F-t graph

• Area under F-t curve = impulse
• Include sign. Area below the axis gives negative impulse.
• Break irregular shapes into triangles and rectangles.

Students often mix this up with momentum graphs. Remember:

• Area under F-t → impulse
• Slope of p-t → force

3. The Impulse-Momentum Theorem

Change in momentum

$$\Delta \mathbf{p} = \mathbf{p}_{f} - \mathbf{p}_{i}$$

It’s vector subtraction. If velocity reverses direction, that’s a large change in momentum.

Core relationship

$$\mathbf{J} = \Delta \mathbf{p}$$

Full version:

$$\int_{t_{1}}^{t_{2}} \mathbf{F}_{\text{net}}(t)\, dt = \Delta \mathbf{p}$$

This is huge for collisions and explosions. You usually don’t know the detailed force function, but you can relate impulse directly to momentum change.

Example setup you’ll see on a test:

• A 0.50 kg cart goes from $+4\text{ m/s}$ to $-2\text{ m/s}$.
• $\Delta p = m(v_{f} - v_{i}) = 0.50(-2 - 4) = -3\text{ kg}\cdot\text{m/s}$.
• Impulse is $-3\text{ N}\cdot\text{s}$.

Negative just means opposite your positive direction.

Newton’s second law as a special case

If mass is constant:

$$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt} = m\frac{d\mathbf{v}}{dt} = m\mathbf{a}$$

So $F = ma$ is just the constant-mass version of the impulse-momentum theorem.

4. Variable-Mass Systems

Start from the general law:

$$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}$$

If velocity is constant but mass changes:

$$\mathbf{F}_{\text{net}} = \frac{d}{dt}(m\mathbf{v}) = \frac{dm}{dt}\mathbf{v}$$

So even with constant velocity, a system can have a net force if mass is flowing in or out.

This shows up in:

• Rocket propulsion
• Fuel ejection
• Systems gaining mass (like falling snow sticking to an object)

On exams, the trap is automatically writing $F = ma$. If mass changes, go back to $F = dp/dt$.

5. Setting Up Impulse Problems

Given a force or F-t graph

1. Find impulse
   • Integrate $F(t)$
   • Or compute area under graph
2. Use $J = \Delta p$
3. Substitute $p = mv$ if mass is constant

Given velocities

1. Compute $\Delta p = m(v_{f} - v_{i})$
2. Set $J = \Delta p$
3. If needed, find $F_{\text{avg}} = \frac{\Delta p}{\Delta t}$

Be careful with signs. Many FRQs deduct for direction errors, not math mistakes.