Describing motion precisely
Kinematics describes how objects move: where they are, how fast they go and how their speed changes. It doesn’t ask why (that’s dynamics).
Scalars and vectors
| Scalar (magnitude only) | Vector (magnitude and direction) |
|---|---|
| Distance: 5 km | Displacement: 5 km east |
| Speed: 20 m/s | Velocity: 20 m/s north |
| Time, mass, temperature | Acceleration, force |
Example: You walk 3 km east and then 3 km west. Distance = 6 km. Displacement = 0 km, because you ended where you started.
Velocity and acceleration
Average velocity = displacement ÷ time: v = Δd ÷ Δt
Acceleration = change in velocity ÷ time: a = Δv ÷ Δt
A negative acceleration doesn’t always mean slowing down. It means acceleration in the negative direction. If an object is moving in the negative direction and has negative acceleration, it’s speeding up.
The kinematic equations
For motion with constant acceleration:
| Equation | Missing variable |
|---|---|
| v_f = v_i + aΔt | Δd |
| Δd = v_iΔt + ½aΔt² | v_f |
| v_f² = v_i² + 2aΔd | Δt |
| Δd = ½(v_i + v_f)Δt | a |
Problem-solving steps
- List known values (with signs) and the unknown.
- Choose the equation that contains them but not the missing variable.
- Solve and check units.
Example: A car accelerates from rest at 4 m/s² for 5 s. How far does it go?
- v_i = 0, a = 4 m/s², Δt = 5 s, Δd = ?
- Δd = 0 + ½ × 4 × 5² = 50 m
Example: A car at 25 m/s brakes at −5 m/s². How far does it travel before stopping?
- v_f² = v_i² + 2aΔd → 0 = 625 + 2(−5)Δd → Δd = 62.5 m
Free fall
In free fall, gravity is the only force. Near Earth’s surface, all objects accelerate downward at g = 9.8 m/s² (ignoring air resistance).
Galileo is said to have dropped balls from the Leaning Tower of Pisa to show that heavy and light objects fall at the same rate. In 1971, Apollo 15 astronaut David Scott dropped a hammer and a feather on the Moon; with no air, they landed together.
Example: A stone is dropped from a 45 m cliff. How long does it take to land?
- Δd = ½gt² → 45 = ½(9.8)t² → t² = 9.18 → t ≈ 3.0 s
Motion graphs
Position–time (d–t) graphs
| Shape | Meaning |
|---|---|
| Horizontal line | At rest |
| Straight sloped line | Constant velocity |
| Curve | Accelerating |
Slope = velocity.
Velocity–time (v–t) graphs
| Shape | Meaning |
|---|---|
| Horizontal line | Constant velocity (a = 0) |
| Straight sloped line | Constant acceleration |
| Line crossing the time axis | Object changes direction |
- Slope = acceleration.
- Area under the line = displacement.
Example: A v–t graph rises from 0 to 20 m/s over 4 s. Acceleration = 20 ÷ 4 = 5 m/s². Displacement = area of triangle = ½ × 4 × 20 = 40 m.
Acceleration–time (a–t) graphs
Area under the line = change in velocity.
Projectile motion (preview)
A thrown ball moves horizontally at constant velocity while accelerating downward due to gravity. The two motions are independent. A bullet fired horizontally and one dropped from the same height hit the ground at the same time (ignoring air resistance).
Common mistakes
- Confusing distance and displacement.
- Forgetting signs. Choose a positive direction and stick to it.
- Reading a v–t graph like a d–t graph.
- Using kinematic equations when acceleration isn’t constant.
Try it yourself
- A cyclist goes from 2 m/s to 10 m/s in 4 s. Find the acceleration and the distance covered.
- A ball is thrown straight up at 14.7 m/s. How long until it reaches the top? How high does it go?
- Sketch a v–t graph for a car that accelerates, cruises, then brakes to a stop.
(Answers: 2 m/s², 24 m; 1.5 s, about 11 m)
Quick recap
- Kinematics describes motion with displacement, velocity and acceleration.
- Vectors have direction; scalars don’t.
- Four kinematic equations apply to constant acceleration.
- Free-fall acceleration is 9.8 m/s² downward.
- On a v–t graph, slope is acceleration and area is displacement.