Dynamics An understanding of forces from Cambridge IGCSE/O Level Physics or equivalent is assumed.
Equations of Motion
1️⃣ Constant Acceleration
When an object moves with a constant acceleration, its motion can be described by four main equations. Think of a car accelerating from rest on a straight road.
| Equation | Meaning |
|---|---|
| $v = u + at$ | Final velocity after time $t$. |
| $s = ut + \tfrac{1}{2}at^2$ | Distance travelled in time $t$. |
| $v^2 = u^2 + 2as$ | Final velocity in terms of distance. |
| $s = \tfrac{1}{2}(u+v)t$ | Distance using average velocity. |
🔍 Tip: Always check units – metres, seconds, metres per second squared.
2️⃣ Variable Acceleration
When acceleration changes, we use calculus: $a(t) = \frac{dv}{dt}$ and $v(t) = \frac{ds}{dt}$. For example, a roller‑coaster’s speed changes as it goes up and down hills.
Block equation for velocity:
$$v(t) = \int a(t)\,dt + C$$Block equation for position:
$$s(t) = \int v(t)\,dt + C$$🛠️ Analogy: Think of a water tap – the flow rate (acceleration) can vary, so the amount of water (distance) changes over time.
3️⃣ Projectile Motion (2D)
Projectiles follow a parabolic path under constant gravitational acceleration $g = 9.81\,\text{m/s}^2$ (downwards).
| Component | Equation |
|---|---|
| Horizontal | $x = v_0 \cos\theta \, t$ |
| Vertical | $y = v_0 \sin\theta \, t - \tfrac{1}{2}gt^2$ |
🚀 Example: Throw a ball at $45^\circ$ with $v_0 = 20\,\text{m/s}$. Calculate its maximum height and range.
4️⃣ Work, Energy & Power
Work done by a constant force: $W = Fd \cos\phi$.
Kinetic energy: $K = \tfrac{1}{2}mv^2$.
Potential energy in a gravitational field: $U = mgh$.
Power (rate of doing work): $P = \frac{W}{t} = Fv$.
💡 Analogy: Pushing a box up a hill – the work is the effort, the energy is stored as height, and power is how quickly you push.
5️⃣ Conservation Laws (Exam Focus)
• Momentum: $\displaystyle \vec{p} = m\vec{v}$; conserved in isolated systems.
• Energy: Total mechanical energy conserved when no non‑conservative forces act.
🔎 Exam Tip: Identify if the problem involves an isolated system (no external forces) before applying conservation.
📚 Examination Tips
1️⃣ Read the question carefully. Note the given variables and what is asked (time, distance, velocity, etc.). 2️⃣ Choose the right equation. Use the variables you have; avoid unnecessary ones. 3️⃣ Check units. Convert all quantities to SI units before calculation. 4️⃣ Show all steps. Even if you know the answer, partial marks are awarded for clear work. 5️⃣ Use diagrams. Sketch a quick diagram for projectile or motion problems – it helps organise your thoughts. 6️⃣ Time management. Allocate 10–15 min for each problem; if stuck, move on and return if time permits. 7️⃣ Review key constants. Remember $g = 9.81\,\text{m/s}^2$, $F = ma$, $W = Fd$, $P = Fv$. 8️⃣ Practice past papers. Familiarise yourself with the style of questions and the marking scheme. 9️⃣ Check your answer. Verify that it makes sense (e.g., a velocity cannot be negative if the direction is specified). 🔟 Stay calm. Confidence and clarity improve accuracy.
💬 Final Thought
Equations of motion are the language of dynamics. By mastering them, you can predict how anything from a falling apple to a spacecraft will move. Keep practising, ask questions, and soon you’ll feel as confident as a rocket scientist! 🚀
Revision
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