The formal definition of acceleration is
It is the measure of how fast the velocity changes with respect to time.
Now, when the magnitude of velocity changes, acceleration $$\vec{a} = \frac{d \vec{v}}{dt}$$ and its unit is $m.s^{-2}$ .
But what about the case where the velocity only changes its direction keeping its magnitude constant? The definition can then be
Acceleration is the measure of how fast the velocity changes its direction with respect to time.
But how can I calculate the acceleration then?? What should be the unit then??? Please help.
Best Answer
The change in velocity can be calculated by vector subtraction. ($d\vec{v} = \vec{v_f} - \vec{v_i}$).
Divide by the time between the two velocities to generate an acceleration. The direction of the acceleration will be the same as the direction of the difference vector. The magnitude of the acceleration will be the same as the magnitude of the difference vector divided by the time.
Example, at time $t=0$ seconds the velocity is in the $x$ direction, and at time $t=2$ seconds the velocity is in the y direction. At both times the velocity is 1 m/s: $$\vec{v(0)} = (1,0,0) ~\mathrm{m/s}$$ $$\vec{v(2)} = (0,1,0) ~\mathrm{m/s}$$ $$d\vec{v} = \vec{v(2)} - \vec{v(0)}$$ $$d\vec{v} = (-1,1,0) ~\mathrm{m/s}$$ $$dt = 2 - 0 = 2 ~\mathrm{s}$$ $$\vec{a} = \frac{d\vec{v}}{dt} = (-\frac12,\frac12,0) ~\mathrm{m/s^2}$$