First slide

Rectilinear Motion

Question

A car, starting from rest, is accelerated at a constant rate $α$ until it attains a speed $ν$ . It is then retarded at a constant rate $β$ until it comes to rest. The average speed of the car during its entire journey is

Moderate

A

zero
B

$\frac{αv}{2 β}$
C

$\frac{βv}{2 α}$
D

$\frac{v}{2}$

Solution

The distance s₁ covered by the car during the time it is accelerated is given by 2 ${αS}_{1}$ = v², which gives $s_{1} = v^{2} / 2 α$ .
The distance s₂ covered during the time the car is decelerated is, similarly given by $s_{2} = v^{2} / 2 β$ .
Therefore, the total distance covered is
$s = s_{1} + s_{2} = \frac{v^{2}}{2} (\frac{1}{α} + \frac{1}{β}) . . . (i)$
If t₁ is the time of acceleration and t₂ that of deceleration, then $v = {αt}_{1} = {βt}_{2} or t_{1} = v / α and t_{2} = v / β$
$t = t_{1} + t_{2} = v (\frac{1}{α} + \frac{1}{β}) . . . (ii)$
From (i) and (ii), the average speed of the car is given by
$\frac{total distance}{total time} = \frac{s}{t} = \frac{v}{2}$ .

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