The complete lecture — same content as the written version, but every concept comes alive in the live panel on the right as you read. Scroll down; the animation keeps pace.
1 — Rest, motion & kinematics
Mechanics is the study of motion. It splits into kinematics — describing motion (distance, speed, acceleration) — and dynamics, which asks why motion changes (forces). This chapter is kinematics.
- Rest — a body is at rest if its position does not change with respect to its surroundings.
- Motion — a body is in motion if its position changes with respect to its surroundings.
- Frame of reference — the surroundings (a coordinate system) against which we measure position. Rest and motion are always relative to a frame.
Relative motion: a passenger sitting in a moving bus is at rest relative to the bus, but in motion relative to the road.
2 — Scalars & vectors
- Scalar — a quantity with magnitude only. distance, speed, time, mass, energy.
- Vector — a quantity with magnitude and direction. displacement, velocity, acceleration, force.
A vector is drawn as an arrow: its length is the magnitude, the arrowhead shows direction. "5 m/s" is a speed; "5 m/s due east" is a velocity.
3 — Distance & displacement
- Distance — total length of the path actually travelled. A scalar, always positive.
- Displacement — the shortest straight-line distance from start to finish, with direction. A vector.
SI unitboth measured in metres (m)
distance ≥ |displacement|, always
Exam point: if you walk all the way around a circular track and return to the start, your distance is the full lap but your displacement is zero.
4 — Speed & velocity
- Speed — rate of change of distance. A scalar. v = distance / time.
- Velocity — rate of change of displacement. A vector. v = displacement / time.
Definitionsaverage speed = total distance / total time
velocity = displacement / time (unit: m/s)
Uniform velocity: equal displacements in equal intervals of time, in a fixed direction.
5 — Acceleration
- Acceleration — rate of change of velocity. A vector, unit m/s².
Accelerationa = (v − u) / t
u = initial velocity · v = final velocity · t = time
Retardation (deceleration) is negative acceleration — velocity decreasing. The dots in the panel bunch closer together.
6 — Distance–time graphs
Plot distance (y-axis) against time (x-axis). The gradient (slope) of the line gives the speed.
| Graph shape | Means |
|---|
| Horizontal line | at rest (speed = 0) |
| Straight sloping line | uniform (constant) speed |
| Curve getting steeper | accelerating |
7 — Velocity–time graphs
Plot velocity against time. Two powerful readings come straight off this graph:
Two key readingsslope of the line = acceleration (a)
area under the line = distance / displacement (S)
area = distance
A body reaches 20 m/s in 10 s from rest.
area = ½ × base × height = ½ × 10 × 20 = 100 m
8 — Equations of motion
For motion in a straight line with uniform acceleration, three equations connect the five quantities (u, v, a, t, S):
The three equations① v = u + a t
② S = u t + ½ a t²
③ 2 a S = v² − u²
Tip: list what you're given and what you want, then pick the equation that contains exactly those letters.
9 — Motion under gravity (free fall)
Near the Earth's surface, every freely-falling body accelerates downward at g ≈ 9.8 m/s² (often taken as 10 m/s² for quick work), regardless of its mass.
Replace a with gv = u + g t
h = u t + ½ g t²
2 g h = v² − u²
dropped from rest
u = 0, t = 3 s, g = 10
v = 0 + 10 × 3 = 30 m/s
h = ½ × 10 × 3² = 45 m
10 — Projectile motion
- Projectile — a body thrown into the air that then moves only under gravity. Its path is a parabola.
The motion splits into two independent parts: a constant horizontal velocity and a vertical motion under gravity.
For launch speed u at angle θ (g down)Time of flight T = 2u sinθ / g
Max height H = u²sin²θ / 2g
Range R = u² sin2θ / g (max at θ = 45°)
Change the sliders in the live panel and press Launch to see the range and height respond.
11 — Worked numericals
Find the distance
A car starts from rest and accelerates at 2 m/s² for 6 s. How far does it travel?
u = 0, a = 2, t = 6
S = ut + ½at² = 0 + ½(2)(6²) = 36 m
Find the final velocity
A train moving at 15 m/s brakes at 3 m/s². What is its speed after 4 s?
u = 15, a = −3, t = 4
v = u + at = 15 + (−3)(4) = 3 m/s
Use the third equation
A bike accelerates from 10 to 20 m/s over 75 m. Find a.
2aS = v² − u² → a = (20² − 10²)/(2×75) = 2 m/s²
12 — Exam recap
- Rest & motion are relative to a frame of reference.
- Scalars (magnitude) vs vectors (magnitude + direction).
- Distance vs displacement; speed vs velocity.
- Acceleration a = (v − u)/t, unit m/s².
- Graphs: slope & area readings.
- Three equations of motion; free fall (g); projectiles.