understand that objects moving against a resistive force may reach a terminal (constant) velocity
Momentum and Newton’s Laws of Motion
1. What is Momentum?
Momentum is a measure of how much motion an object has. It is the product of mass and velocity: $p = m\,v$. Think of it as the “push” an object carries when it moves. 🌟 Analogy: Imagine a bowling ball rolling down a hallway. The heavier the ball or the faster it rolls, the more momentum it has, so it’s harder to stop.
2. Newton’s Laws of Motion
- First Law (Inertia): An object stays at rest or moves at constant speed unless a net external force acts on it. ⚖️
- Second Law (F = ma): The net force on an object equals its mass times its acceleration: $F_{\text{net}} = m\,a$. This links force, mass, and acceleration. 🚀
- Third Law (Action–Reaction): For every action, there is an equal and opposite reaction. If you push a wall, the wall pushes back on you with the same force. 🤝
3. Resistive Forces and Terminal Velocity
When an object moves through a fluid (air, water), it experiences a resistive (drag) force that grows with speed. The drag force can be approximated by: $F_{\text{drag}} = \tfrac{1}{2}\,\rho\,C_{\text{d}}\,A\,v^{2}$, where $\rho$ is fluid density, $C_{\text{d}}$ is the drag coefficient, $A$ is the cross‑sectional area, and $v$ is velocity. As the object speeds up, $F_{\text{drag}}$ increases until it balances the weight $mg$. At that point, acceleration stops and the object falls at a constant speed – the terminal velocity**.**
Key Insight: Terminal velocity occurs when the net force is zero: $mg = \tfrac{1}{2}\,\rho\,C_{\text{d}}\,A\,v_{\text{t}}^{2}$. Solving for $v_{\text{t}}$ gives the speed at which the object stops accelerating.
4. Everyday Examples
- 🪂 Skydiver: A skydiver initially accelerates, then reaches a constant free‑fall speed (~55 m/s) when drag balances weight.
- 🍃 Falling leaf: Leaves fall slowly because their small mass and large surface area create high drag, so terminal velocity is low.
- 🚗 Car braking: The friction between tires and road provides a resistive force that brings the car to a stop.
5. Quick Practice Problems
- Calculate the momentum of a 2 kg ball moving at 10 m/s.
- If a 70 kg skydiver has a drag coefficient of 1.0 and a cross‑sectional area of 0.7 m², estimate the terminal velocity in air (density ≈ 1.2 kg/m³).
- Explain why a heavier object (same shape) reaches a higher terminal velocity than a lighter one.
6. Summary Table
| Property | Formula / Example |
|---|---|
| Momentum | $p = m\,v$ (e.g., 2 kg × 10 m/s = 20 kg·m/s) |
| Drag Force | $F_{\text{drag}} = \tfrac{1}{2}\,\rho\,C_{\text{d}}\,A\,v^{2}$ |
| Terminal Velocity | $v_{\text{t}} = \sqrt{\dfrac{2mg}{\rho\,C_{\text{d}}\,A}}$ |
🎉 Remember: When forces balance, motion becomes steady. Understanding how resistive forces work helps predict real‑world behaviours, from falling leaves to skydiving adventures!
Revision
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