## Question

A bus accelerates from the rest for time t1 at a constant rate α then retards at a constant rate β for time t2 and comes to rest, Then t1/t2

a. α/(β+α)

b. (β+α)/2

c. (β+α)/α

d. β/α

### Solution

To solve for the ratio t_{1}/t_{2}, we will analyze the motion of the bus in two phases: acceleration and deceleration.

#### Phase 1: Acceleration

- The bus starts from rest and accelerates at a constant rate α for time t
_{1}. - The final velocity at the end of this phase, v
_{1}, is given by: v_{1}= αt_{1}

#### Phase 2: Deceleration

- The bus then decelerates at a constant rate β for time t
_{2} and comes to rest. - The initial velocity for this phase is v
_{1}, and the final velocity is 0. - Using the equation of motion for constant deceleration: 0 = v
_{1}−βt_{2}

Substituting v_{1} from the first phase: 0 = αt_{1 }− βt_{2}

Solving for: t_{1 }/ t_{2}

Rearranging the equation: αt_{1 }= βt_{2}

Dividing both sides by βt_{2}, we will now have (αt_{1 }/ βt_{2}) = 1

Thus, t_{1} / t_{2}= β / α

**Therefore, the correct answer is: d. β/α**