Summary — What This Chapter Covers
- Distance and speed are scalars; displacement, velocity and acceleration are vectors.
- Gradient of displacement–time = velocity; gradient of velocity–time = acceleration; area under velocity–time = displacement.
- The equations of uniformly accelerated motion are derived from the definitions of velocity and acceleration.
- Free fall near the Earth: a = g = 9.81 m s−2 downwards (no air resistance).
- g can be measured by timing a falling object over known distances.
- A projectile has constant horizontal velocity and constant vertical acceleration g; treat the two directions separately.
1. Describing Motion and Motion Graphs
Equations of motion
Definitions, gradients and areas
Displacement is distance in a stated direction. Speed = distance/time; velocity = change in displacement per unit time (rate of change of displacement); acceleration = change in velocity per unit time (rate of change of velocity).
On a displacement–time graph the gradient is the velocity. On a velocity–time graph the gradient is the acceleration and the area between the line and the time axis is the displacement (area below the axis counts as negative displacement). For a curve, draw a tangent to find an instantaneous gradient.
Definitions
v = Δs/Δt a = Δv/Δt
Worked Example
A car accelerates uniformly from rest to 12 m s−1 in 4.0 s, travels at 12 m s−1 for 6.0 s, then decelerates uniformly to rest in 3.0 s. Find the acceleration, the deceleration and the total distance.
2. Equations of Uniformly Accelerated Motion
Equations of motion
suvat, derived and used
For constant acceleration a, from a = (v − u)/t we get v = u + at. The average velocity is (u + v)/2, so s = ½(u + v)t. Substituting v = u + at gives s = ut + ½at2; eliminating t gives v2 = u2 + 2as.
Choose a positive direction first and give every vector (s, u, v, a) a sign. For an object thrown upwards with "up" positive, a = −9.81 m s−2 for the whole flight, including at the top where v = 0.
Uniformly accelerated motion
v = u + at s = ½(u + v)t s = ut + ½at2 v2 = u2 + 2as
Worked Example
A stone is dropped from rest from a cliff 45.0 m high. Ignoring air resistance, find the time to reach the sea and its speed on impact.
Practice Question
A ball is thrown vertically upwards at 15.0 m s−1. Find its maximum height and the time it takes to return to the thrower.
Experiment: measuring g
Release a steel ball from an electromagnet so that the timer starts as the current is switched off; the ball breaks a light gate (or opens a trapdoor) a measured height h below, stopping the timer. Repeat for several heights. Since h = ½gt2, plot h against t2: a straight line through the origin with gradient g/2. A gradient of 4.90 m s−2 gives g = 9.80 m s−2. Measure h to the bottom of the ball, and repeat each time to reduce random error.
3. Motion in Two Perpendicular Directions
Equations of motion
Projectiles
uniform velocity in one direction