Topic 1 of 6
Describe motion with a reference
Choose an origin, a positive direction and an interval. Then distinguish where a body is, how far it has travelled and how its velocity is changing.
- Position
- Location relative to a chosen origin. In one dimension, x = +2 m means 2 m in the positive direction from the origin.
- Distance travelled
- The accumulated length of the path. It is a scalar, is nonnegative and cannot decrease as the journey continues.
- Displacement
- The directed change in position. For an interval, s = xfinal - xinitial. It can be positive, negative or zero in a chosen coordinate direction.
- Speed and velocity
- Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time. Instantaneous velocity describes the local rate of change of position; instantaneous speed is its magnitude.
- Acceleration
- The rate of change of velocity. Average acceleration is Δv/Δt; instantaneous acceleration is the local rate at a particular time. A change in direction can produce acceleration even when speed is constant.
Distance and displacement are measured in m; speed and velocity in m/s; acceleration in m/s2. A negative position does not by itself mean leftward motion. Position tells you where the body is; velocity tells you how that position is changing.
A velocity change is a vector difference: final velocity minus initial velocity. A change in speed alone does not include every possible change in velocity.
Follow one motion through a reversal
Choose right as positive. At t = 0, a body is at x = 2.0 m with velocity +6.0 m/s. Its acceleration is constant at -2.0 m/s2 for the next 5.0 s.
Equal time intervals do not mean equal distances
The two rows share the same position scale. Dots are 1 s apart in time; the return row repeats the turning point so you can follow the order.
At 3 s, velocity is zero but acceleration is not. At 4 s the position is still positive while the body is moving left. Over the full 5 s, displacement is +5 m and distance is 13 m.
| Time / s | Position / m | Velocity / m s-1 |
|---|---|---|
| 0 | 2 | +6 |
| 1 | 7 | +4 |
| 2 | 10 | +2 |
| 3 | 11 | 0 |
| 4 | 10 | -2 |
| 5 | 7 | -4 |
At 3 s, the body reaches x = 11 m and is instantaneously at rest. Its acceleration is still -2 m/s2, so it reverses rather than staying at rest. At 4 s, it is still to the right of the origin, but its negative velocity means it is moving left.
Worked averages
Distance and displacement answer different questions
The body moves from x = 2 m to 11 m, then back to 7 m.
Distance = (11 - 2) + (11 - 7) = 13 m.
Displacement = 7 - 2 = +5 m.
Over 5.0 s, average speed = 13/5 = 2.6 m/s, while average velocity = 5/5 = +1.0 m/s.
Use both signs to decide whether speed increases
In one dimension, velocity and acceleration of opposite signs mean decreasing speed. Equal signs mean increasing speed. In this model, negative acceleration first slows rightward motion, then speeds up leftward motion after the turn.
Negative acceleration does not always mean slowing down. Acceleration tells you how velocity changes, not the direction in which the body is already moving.