First slide

Rectilinear Motion

Question

A car accelerates from rest at a constant rate $α$ for some time, after which it decelerates at a constant rate $β$ and comes to rest. If the total time elapsed is t, then the maximum velocity acquired by the car is

Moderate

A

$(\frac{α^{2} + β^{2}}{α β}) t$
B

$(\frac{α^{2} - β^{2}}{α β}) t$
C

$\frac{(α - β) t}{α β}$
D

$\frac{α β t}{α + β}$

Solution

Let the car accelerate at rate $α$ for time t₁ Then maximum velocity attained, $v = 0 + α t_{1} + α t_{1}$
Now, the car decelerates at a rate $β$ for time (t - t₁) and finally comes to rest. Then,
$\begin{array}{l} 0 = v β (t - t_{1}) \Rightarrow 0 = α t_{1} - β t + β t_{1} \\ \Rightarrow t_{1} = \frac{β}{α + β} t \\ ∴ v = \frac{α β}{α + β} t \end{array}$

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