# Question a7378

Oct 24, 2016

(b)

#### Explanation:

The car accelerates from rest with an acceleration of  alpha# for a time ${t}_{1}$.
velocity after time ${t}_{1}$ is given by the expression
$v = u + a t$
Inserting given values we have
$v = 0 + \alpha {t}_{1}$
$\implies v = \alpha {t}_{1}$ .......(1)
Thereafter it is decelerated at a rate $\beta$ for a time ${t}_{2}$ when it comes to halt.
We have
$0 = v - \beta {t}_{2}$
$\implies v = \beta {t}_{2}$ .....(2)
Total time ${t}_{1} + {t}_{2} = t$
$\implies {t}_{2} = t - {t}_{1}$
Inserting in (2) we get
$v = \beta \left(t - {t}_{1}\right)$ ......(3)
From (1) we have ${t}_{1} = \frac{v}{\alpha}$. Inserting in (3) above
$v = \beta \left(t - \frac{v}{\alpha}\right)$

Solving for $v$, we have

$v + \frac{\beta}{\alpha} v = \beta t$
$\implies v \left(1 + \frac{\beta}{\alpha}\right) = \beta t$
$\implies v = \frac{\beta t}{1 + \frac{\beta}{\alpha}}$
$\implies v = \frac{\alpha \beta t}{\alpha + \beta}$