Question

A car accelerates from rest at a constant rate of $\alpha$ for some time, after which it decelerates at a constant rate of $\beta$ to come to rest. If the total time elapsed is t second, then calculate
a. the maximum velocity attained by the car, and
b.the total displacement travelled by car in terms of $\alpha ,\beta$ and t.

Answer 1

(a) $\Rightarrow V_0=0+\alpha \left(t_1\right)\Rightarrow t_1=V_0/\alpha$

$\Rightarrow 0=V_0-\beta (t-t_1)$

$\Rightarrow V_0=\beta (t-\frac{V_0}{\alpha })$

$\Rightarrow V_0=\beta t-\frac{\beta V_0}{\alpha }$

$\Rightarrow (1+\frac{\beta }{\alpha })V_0=\beta t$

$\Rightarrow V_0=\frac{\beta t}{(1+\frac{\beta }{\alpha })}$

(b) Total displacement $=$ area of graph -

$=\frac{1}{2}\times V_0\times t$

$=\frac{t}{2}\left[\frac{\beta t}{1+\beta /\alpha }\right]$