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Acceleration

A train accel...

Question

A train accelerates from rest for time $t_{1}$ at a constant rate $a$ and then it retards at the constant rate $b$ for time $t_{2}$ and comes to rest. Find the ratio $\frac{t_{1}}{t_{2}}$ .

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Solution

Initial velocity $u$ = $0$
Accelerates with acceleration $a$ for $t_{1}$ and reaches maximum velocity $v$
So
$v = u + a t$
$v = a t_{1}$
Then retards from maximum speed with negative acceleration $b$ for time $t_{2}$
So
$0 = v - b t_{2}$ because final velocity is zero.
$v = b t_{2}$
$a t_{1} = b t_{2}$
$\frac{t_{1}}{t_{2}}$ = $\frac{b}{a}$

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