define and use force as rate of change of momentum
Momentum and Newton’s Laws of Motion
What is Momentum?
Momentum is a measure of how much motion an object has. It is the product of an object’s mass and its velocity:
$$p = mv$$
Think of a soccer ball: the heavier the ball and the faster it rolls, the more momentum it carries.
Newton’s Second Law: Force as Rate of Change of Momentum
Newton’s second law can be written in a form that shows force as the rate at which momentum changes:
$$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$
In words: the force acting on an object is equal to the time‑rate of change of its momentum.
Analogy: The Car on a Highway 🚗
Imagine a car traveling at a constant speed. Its momentum is steady. If you suddenly press the gas pedal, the car speeds up. The engine provides a force that changes the car’s momentum. The faster you press the pedal, the larger the force, and the quicker the momentum changes.
Examples
- Throwing a ball: A 0.5 kg ball is thrown at 10 m s⁻¹. Its momentum is $p = 0.5 \times 10 = 5 \,\text{kg·m s}^{-1}$. If the ball’s speed changes to 15 m s⁻¹ in 0.2 s, the average force applied is $$\mathbf{F} = \frac{\Delta p}{\Delta t} = \frac{(0.5 \times 15) - (0.5 \times 10)}{0.2} = \frac{7.5 - 5}{0.2} = 12.5 \,\text{N}.$$
- Stopping a moving train: A 200 000 kg train traveling at 20 m s⁻¹ is brought to a halt in 10 s. The average braking force is $$\mathbf{F} = \frac{0 - (200\,000 \times 20)}{10} = -400\,000 \,\text{N}.$$ The negative sign indicates the force opposes the motion.
Key Concepts in a Box
- Momentum ($p$) = mass ($m$) × velocity ($v$).
- Force ($F$) = rate of change of momentum ($dp/dt$).
- Units: $p$ in kg·m s⁻¹, $F$ in N (kg·m s⁻²).
- Direction of $F$ is the direction in which momentum changes.
Exam Tips for A-Level Physics
🔍 Understand the formula: $F = dp/dt$ is key for many questions. Remember that $p = mv$ if mass is constant.
📐 Units matter: Always check that your final answer has the correct SI units.
🧮 Use the right sign: Positive or negative forces indicate direction. Pay attention to the problem’s coordinate system.
📝 Show all steps: Even if you know the answer, write out the intermediate steps. Mark your work clearly.
💡 Check assumptions: Is mass constant? Is the motion one‑dimensional? Clarify these before applying the formula.
Units Table
| Quantity | Symbol | Unit |
|---|---|---|
| Mass | $m$ | kg |
| Velocity | $v$ | m s⁻¹ |
| Momentum | $p$ | kg·m s⁻¹ |
| Force | $F$ | N (kg·m s⁻²) |
Revision
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