Parallel and Perpendicular Components of Acceleration

The acceleration of a particle moving in a curved path can be represented in terms of components parallel and perpendicular to the velocity at each point. In the following figure, these components are labeled apar and aperp. To see why these components are useful, two special cases will be considered. In the figure, the acceleration vector is parallel to the velocity

v
1. The change in

v
during a small time interval Δt is a vector Δ

v
having the same direction as a and hence the same direction as

v
1. The velocity

v
2 at the end of Δt, given by

v
2 =

v
1 + Δ

v
, is a vector having the same direction as

v
1 but greater magnitude. In other words, during the time interval Δt the particle in the figure moved in a straight line with increasing speed.

In the figure, the acceleration

a
is perpendicular to the velocity

v
. In a small time interval Δt, the change Δv is a vector very nearly perpendicular to

v
1, as shown. Again

v
2 =

v
1 + Δ

v
, but in this case

v
1 and

v
2 have different directions. As the time interval Δt approaches zero, the angle φ in the figure also approaches zero, Δ

v
becomes perpendicular to both

v
1 and

v
2, and

v
1 and

v
2 have the same magnitude. In other words, the speed of the particle stays the same, but the path of the particle curves.

Thus when

a
is parallel (or antiparallel) to

v
, its effect is to change the magnitude of v but not its direction; when

a
is perpendicular to

v
, its effect is to change the direction of

v
but not its magnitude. In general,

a
may have components both parallel and perpendicular to

v
, but the above statements are still valid for the individual components. In particular, when a particle travels along a curved path with constant speed, its acceleration is always perpendicular to

v
at each point.

The figure shows a particle moving along a curved path for three different situations: constant speed, increasing speed, and decreasing speed. If the speed is constant, a is perpendicular, or normal, to the path and to

v
and points toward the concave side of the path, as in the figure. If the speed is increasing, there is still a perpendicular component of

a
, but there is also a parallel component having the same direction as

v
. Then

a
points ahead of the normal to the path. If the speed is decreasing, the parallel component has the direction opposite to

v
, and

a
points behind the normal to the path, as in the figure. These ideas are used in the lessons Uniform Circular Motion and Non-Uniform Circular Motion.

Note that the two quantities:

d|v|
dt

and

dv
dt

are not the same. The first is the rate of change of speed; it is zero whenever a particle moves with constant speed, even when its direction of motion changes. The second is the magnitude of the vector acceleration; it is zero only when the particle's acceleration is zero, that is, when the particle moves in a straight line with constant speed.