a bus accelerates from rest for time t1

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 t1/t2, 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 t1​.
• The final velocity at the end of this phase, v1​, is given by: v1 = αt1

Phase 2: Deceleration

• The bus then decelerates at a constant rate β for time t2​ and comes to rest.
• The initial velocity for this phase is v1​, and the final velocity is 0.
• Using the equation of motion for constant deceleration: 0 = v1 −βt2

Substituting v1​ from the first phase: 0 = αt1 ​− βt2

​Solving for: t1 ​/ t2

Rearranging the equation: αt1 ​= βt2

​Dividing both sides by βt2, we will now have (αt1 ​/ βt2) = 1

Thus, t1 / t2​= β / α

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