Dynamics An understanding of forces from Cambridge IGCSE/O Level Physics or equivalent is assumed.

Equations of Motion

In dynamics we describe the motion of a particle using its displacement, velocity and acceleration. The following notes assume you are already familiar with the concepts of force, mass and Newton’s second law.

1. Kinematic \cdot ariables

• Displacement ($s$) – change in position measured along a straight line.
• Velocity ($v$) – rate of change of displacement; $v = \dfrac{ds}{dt}$.
• Acceleration ($a$) – rate of change of velocity; $a = \dfrac{dv}{dt}$.
• Time ($t$) – the independent variable.

2. The Three Core Equations of Motion (Constant Acceleration)

When acceleration is constant, the motion can be described by three equations that relate $s$, $v$, $a$ and $t$.

Equation	Form	When to use
First equation	$$v = u + at$$	Known: initial velocity $u$, acceleration $a$, time $t$.
Second equation	$$s = ut + \frac{1}{2}at^{2}$$	Known: $u$, $a$, $t$; need displacement $s$.
Third equation	$$v^{2} = u^{2} + 2as$$	Known: $u$, $a$, $s$; need final velocity $v$ (no time required).

3. Derivation of the Third Equation

Starting from the first two equations:

$$v = u + at \quad\text{and}\quad s = ut + \frac{1}{2}at^{2}$$

Eliminate $t$ by solving the first for $t = \dfrac{v-u}{a}$ and substituting into the second:

$$s = u\left(\frac{v-u}{a}\right) + \frac{1}{2}a\left(\frac{v-u}{a}\right)^{2}$$

After simplification you obtain:

$$v^{2} = u^{2} + 2as$$

4. Sign Conventions

• Choose a positive direction (e.g., to the right or upwards).
• Displacements, velocities and accelerations in the chosen direction are positive; opposite direction are negative.
• Consistent sign usage is essential when solving problems involving multiple forces.

5. Common Applications

1. Free fall – $a = g = 9.81\ \text{m s}^{-2}$ downwards.
2. Projectile motion (horizontal component) – $a = 0$ horizontally, so $v_{x}=u_{x}$ is constant.
3. Uniformly accelerated motion on an inclined plane – $a = g\sin\theta$ (down the slope).

6. Worked Example

Problem: A car accelerates uniformly from rest to $30\ \text{m s}^{-1}$ in $10\ \text{s}$. Find the acceleration and the distance travelled.

Solution:

1. Use the first equation: $v = u + at \;\Rightarrow\; 30 = 0 + a(10)$, so $a = 3\ \text{m s}^{-2}$.
2. Use the second equation: $s = ut + \frac{1}{2}at^{2} = 0 + \frac{1}{2}(3)(10)^{2} = 150\ \text{m}$.

7. Practice Questions

1. A stone is dropped from rest from a cliff 80 m high. Ignoring air resistance, calculate the time taken to reach the ground and its impact speed. (Take $g = 9.81\ \text{m s}^{-2}$.)
2. A projectile is launched horizontally with a speed of $20\ \text{m s}^{-1}$ from a cliff 45 m high. Determine the time of flight and the horizontal distance travelled before it hits the water.
3. A block slides down a frictionless plane inclined at $30^{\circ}$ from rest. Find the speed of the block after it has travelled $5\ \text{m}$ along the plane.
4. A car traveling at $25\ \text{m s}^{-1}$ brakes uniformly to a stop in $8\ \text{s}$. Compute the magnitude of the deceleration and the distance covered during braking.

8. Summary

• The three equations of motion apply only when acceleration is constant.
• Choose a consistent sign convention before solving any problem.
• Use the first equation when time is known, the second when displacement is required, and the third when time is not needed.
• Remember to convert units where necessary and to keep significant figures appropriate for A‑Level examinations.