Describing motion
An object is in motion when its position changes with time, compared with a reference point. A tree is at rest relative to the ground but in motion relative to a passing train.
Distance and displacement
Distance is the total length of the path travelled. It has magnitude only, so it is a scalar.
Displacement is the shortest straight-line distance from the starting point to the final position, in a given direction. It has magnitude and direction, so it is a vector.
Example. You walk 4 m east and then 3 m north. The distance you covered is 4 + 3 = 7 m. Your displacement is √(4² + 3²) = 5 m, pointing north-east of the start.
If you walk round a circular track and come back to the start, the distance is the length of the track, but the displacement is zero.
Speed and velocity
Speed = distance ÷ time. Velocity = displacement ÷ time.
Both are measured in m/s. Speed is a scalar; velocity is a vector.
Uniform motion: equal distances are covered in equal intervals of time.
Average speed = total distance ÷ total time.
Example. A car covers 60 km in the first hour and 40 km in the next hour. Its average speed is (60 + 40) ÷ 2 = 50 km/h.
To convert km/h to m/s, multiply by 5/18. For example, 72 km/h = 72 × 5/18 = 20 m/s.
Acceleration
Acceleration is the rate of change of velocity:
a = (v − u) / t
Here u is the initial velocity, v the final velocity and t the time taken. The unit is m/s².
If the velocity increases, the acceleration is positive.
If the velocity decreases, the acceleration is negative. This is called retardation or deceleration.
Example. A bus speeds up from rest to 10 m/s in 5 s. Its acceleration is (10 − 0) ÷ 5 = 2 m/s².
Graphs of motion
Distance–time graph
A straight sloping line means uniform speed. The slope of the line gives the speed.
A line parallel to the time axis means the object is at rest.
A curved line means non-uniform speed.
Velocity–time graph
A horizontal line means constant velocity (zero acceleration).
A straight sloping line means uniform acceleration. The slope gives the acceleration.
The area under the graph gives the displacement.
The equations of motion
For motion in a straight line with uniform acceleration:
First equation: v = u + at
Second equation: s = ut + ½at²
Third equation: v² = u² + 2as
where s is the displacement.
Example. A car travelling at 20 m/s brakes uniformly and stops in 4 s. Find the acceleration and the stopping distance.
a = (0 − 20) ÷ 4 = −5 m/s²
Using v² = u² + 2as: 0 = 400 + 2(−5)s, so s = 40 m.
Uniform circular motion
An object moving in a circle at constant speed is still accelerating, because the direction of its velocity keeps changing. This is called uniform circular motion. Examples include the tip of a clock's hand, an athlete running round a circular track at steady speed, and (approximately) the Moon going round the Earth.
For an object of speed v moving once round a circle of radius r in time t:
v = 2πr / t