Acceleration and motion
Acceleration vector and rectilinear motion
Why a vector?
Velocity and acceleration are not simply numbers: they are vectors, that is, quantities that have a magnitude (a norm) AND a direction. The acceleration vector indicates the direction in which the velocity is changing.
Three cases in straight-line motion
In straight-line motion, we compare the direction of the acceleration vector with that of the velocity vector:
| Case | Acceleration vector | Effect on motion |
|---|---|---|
| Same direction as velocity | positive (in the direction of motion) | the motion accelerates |
| Opposite direction to velocity | negative (opposite to motion) | the motion slows down (decelerates) |
| Acceleration vector zero | a = 0 | uniform linear motion (constant velocity) |
Practical example
A skier is skiing down a slope: gravity creates acceleration in the direction of the descent, so the skier accelerates. If they brake by using the ‘snowplough’ technique, they create an opposing force: the resultant acceleration may reverse and the skier slows down.
Uniformly Accelerated Rectilinear Motion (UARM)
When the acceleration remains constant over time, we refer to UMA. This is, as a good approximation, the case for free fall near the ground (acceleration g = 9.8 m/s², directed downwards) or a high-speed train (TGV) setting off with a constant tractive force.
A common pitfall to avoid
Never say that an object is ‘accelerating upwards’ simply because it is moving upwards: what matters is the direction of the acceleration vector relative to the velocity vector, not the geometric direction (upwards, downwards) of the motion itself. A ball thrown upwards slows down as it rises, because its acceleration (due to gravity, downwards) is opposite to its velocity (upwards).