Car going round a bend

Problem

Vectors and magnitudes. A car enters a bend at $20\;\text{m/s}$ and leaves it at $20\;\text{m/s}$. Discuss in five lines whether the car accelerated while going round the bend, and in which direction.

Solution

The magnitude of the velocity does not change: it enters and leaves at $20\;\text{m/s}$. But velocity is a vector, and along the bend its direction changes continuously (at entry it points in one direction, at exit in another).

Since the acceleration measures the change of the velocity vector

$\vec a = \frac{\Delta \vec v}{\Delta t}$

and $\Delta \vec v = \vec v_\text{out} - \vec v_\text{in} \neq \vec 0$ (the two vectors have different directions), the car did accelerate, even though the speedometer always read $20\;\text{m/s}$.

This acceleration is directed towards the centre of the bend (centripetal acceleration) and is perpendicular to the velocity. Its magnitude, for circular motion of radius $r$, is:

$a_c = \frac{v^2}{r}$

$\ev{\text{yes, centripetal acceleration towards the centre, magnitude } a_c = v^2/r}$

Collegamenti

Argomenti: Kinematics · Vectors Concetti: Centripetal acceleration · Acceleration