Describing motion in one dimension
Kinematics describes motion without asking what causes it. In one dimension we pick a positive direction and use signs for the rest. Position x says where an object is, displacement Δx is the change in position, and velocity is displacement per unit time. Acceleration is the rate at which velocity changes: a = Δv / Δt. Average speed is total distance divided by total time, while average velocity is displacement divided by time. If you walk away and come back, your average velocity is zero even though your average speed is not.
The constant-acceleration equations
When acceleration is constant, three equations link the five quantities initial velocity v₀, final velocity v, acceleration a, time t and displacement Δx:
- v = v₀ + a t
- Δx = v₀ t + ½ a t²
- v² = v₀² + 2 a Δx
Choose the equation that contains the three known quantities and the one unknown. List what you know first, and keep signs consistent. A braking car has a negative acceleration if forward is positive.
Free fall
Near the Earth's surface, with air resistance ignored, every falling object has the same acceleration of g = 9.8 m/s² downward. This holds on the way up, at the top and on the way down. At the very top of a throw the velocity is zero for an instant but the acceleration is still 9.8 m/s² downward. A ball thrown up at speed v₀ rises for a time v₀ / g, reaches a height v₀² / (2g), and returns to the launch height at speed v₀ after a total time 2 v₀ / g.
Reading graphs
On a position-time graph, slope is velocity. On a velocity-time graph, slope is acceleration and the area between the line and the time axis is displacement. A straight sloped line on a velocity-time graph means constant acceleration.
Worked example
A scooter moving at 4.0 m/s speeds up at 1.5 m/s² for 6.0 s.
- Final speed: v = 4.0 + 1.5 × 6.0 = 13.0 m/s.
- Distance: Δx = 4.0 × 6.0 + ½ × 1.5 × 6.0² = 24 + 27 = 51 m.
- Check with the third equation: v² = 4.0² + 2 × 1.5 × 51 = 16 + 153 = 169, and 13.0² = 169. It matches.
A stone dropped from rest through 20 m takes t = √(2 × 20 / 9.8) = 2.02 s and lands at speed 9.8 × 2.02 = 19.8 m/s.
Common mistakes
- Using the formulas when acceleration is not constant.
- Treating the speed at the top of a throw, which is zero, as if the acceleration were zero too.
- Forgetting the factor ½ in the displacement equation, or the 2 in v² = v₀² + 2 a Δx.
- Mixing up distance with displacement.