Cambridge A-Level Physics 9702 – Equations of Motion
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:
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
Free fall – \$a = g = 9.81\ \text{m s}^{-2}\$ downwards.
Projectile motion (horizontal component) – \$a = 0\$ horizontally, so \$v{x}=u{x}\$ is constant.
Uniformly accelerated motion on an inclined plane – \$a = g\sin\theta\$ (down the slope).
Suggested diagram: A block sliding down a frictionless incline of angle \$\theta\$, showing \$u\$, \$v\$, \$a=g\sin\theta\$, and displacement \$s\$ along the plane.
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:
Use the first equation: \$v = u + at \;\Rightarrow\; 30 = 0 + a(10)\$, so \$a = 3\ \text{m s}^{-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
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}\$.)
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.
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.
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.