define and use force as rate of change of momentum

Momentum and Newton’s Laws of Motion

Learning Objective

Define force as the rate of change of momentum and apply this definition to solve quantitative problems.

1. Momentum – the Fundamental Quantity

The linear momentum $\mathbf{p}$ of a particle of mass $m$ moving with velocity $\mathbf{v}$ is defined as

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

Momentum is a vector quantity; its direction is the same as the velocity vector.

2. Newton’s Second Law in Momentum Form

Newton’s second law can be written most generally as

$$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$

where $\mathbf{F}$ is the net external force acting on the particle.

2.1 Derivation for Constant Mass

If the mass $m$ is constant, the derivative expands to

$$\mathbf{F}= \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt}=m\mathbf{a}$$

Thus the familiar form $\mathbf{F}=m\mathbf{a}$ is a special case of the more general momentum form.

2.2 \cdot ariable‑Mass Systems (e.g., Rockets)

When mass changes with time, the full derivative must be retained:

$$\mathbf{F}= \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt}+\mathbf{v}\frac{dm}{dt}$$

This expression is essential for analysing rockets and other systems where mass is ejected or accreted.

3. Force as the Rate of Change of Momentum

From the definition $\mathbf{F}=d\mathbf{p}/dt$, several useful results follow.

3.1 Impulse–Momentum Theorem

Integrating the definition over a finite time interval $[t_1,t_2]$ gives

$$\int_{t_1}^{t_2}\mathbf{F}\,dt = \mathbf{p}(t_2)-\mathbf{p}(t_1)=\Delta\mathbf{p}$$

The left‑hand side is the impulse $\mathbf{J}$ delivered to the object:

$$\mathbf{J}= \int_{t_1}^{t_2}\mathbf{F}\,dt = \Delta\mathbf{p}$$

Impulse has the same units as momentum (kg·m·s⁻¹) and provides a convenient way to relate forces that act over short time intervals (e.g., collisions).

3.2 Constant Force Example

If a constant force $\mathbf{F}$ acts for a time $\Delta t$, the change in momentum is simply

$$\Delta\mathbf{p}= \mathbf{F}\,\Delta t$$

and the final velocity can be found from $\mathbf{p}=m\mathbf{v}$.

4. Practical Applications

• Predicting the speed of a car after a known thrust is applied for a given time.
• Analyzing collisions using impulse to determine post‑collision velocities.
• Designing rocket propulsion systems by accounting for the $\mathbf{v}\,dm/dt$ term.

5. Worked Example

Problem: A 0.150 kg ball moving at $8.0\ \text{m s}^{-1}$ collides head‑on with a 0.250 kg ball at rest. The collision lasts $0.020\ \text{s}$ and the average force on each ball during the impact is $120\ \text{N}$. Find the final velocities of both balls assuming the collision is perfectly elastic.

1. Use the impulse–momentum theorem for each ball: $$\mathbf{J}= \mathbf{F}\Delta t = \Delta\mathbf{p}$$
2. Write the momentum change for ball 1 (mass $m_1$): $$120\ \text{N}\times0.020\ \text{s}=m_1(v_{1f}-8.0)$$
3. Write the momentum change for ball 2 (mass $m_2$): $$-120\ \text{N}\times0.020\ \text{s}=m_2(v_{2f}-0)$$ (negative sign because the force on ball 2 is opposite in direction.)
4. Solve the two equations for $v_{1f}$ and $v_{2f}$, then verify the elastic‑collision condition $$\frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2=\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2.$$

6. Summary Table of Key Relations

Quantity	Symbol	Definition / Equation	SI Unit
Momentum	$\mathbf{p}$	$\mathbf{p}=m\mathbf{v}$	kg·m·s⁻¹
Force (general)	$\mathbf{F}$	$\mathbf{F}=d\mathbf{p}/dt$	N (kg·m·s⁻²)
Impulse	$\mathbf{J}$	$\mathbf{J}= \int\mathbf{F}\,dt = \Delta\mathbf{p}$	kg·m·s⁻¹ (same as momentum)
Constant‑force momentum change		$\Delta\mathbf{p}= \mathbf{F}\Delta t$
Variable‑mass force		$\mathbf{F}=m\mathbf{a}+\mathbf{v}\,dm/dt$

7. Suggested Diagram

8. Quick Checklist for Solving Problems

• Identify whether mass is constant or changing.
• Write the appropriate form of $\mathbf{F}=d\mathbf{p}/dt$.
• If a time interval is given, use the impulse–momentum theorem.
• Apply conservation of momentum (and kinetic energy for elastic collisions) where required.
• Check units and sign conventions (direction of vectors).