Question

A body starts from rest and travels a
distance S with uniform acceleration, then moves
uniformly a distance 2S and finally comes to rest after
moving further 5S under uniform retardation. Find the
ratio of average velocity to maximum velocity :​

Answer 1

Answer:

Let a be the uniform acceleration.

  v² - u² = 2 a S    =>  S = v² / 2 a        as u = 0.

  final velocity =  v = √(2 a S)

  time taken to reach this velocity :  t1 = (v - u ) / a  = v / a = √(2S/a)

Now the particle moves with this velocity for distance 2 S.

   so time taken t2 = 2 S / v = 2 S / √(2aS) =  √(2 S / a)

Now the particle moves with a uniform deceleration  - a3 for a distance 3 S.

     time taken t3 to stop.

           2 * a3 * 3 S = v² - u²         here v = 0  and u = √(2 a S)

           =>  a3 =  a / 3 

            =>  t3 = - u /a3 =  3 √( 2 S / a)

total time duration =  t = t1 + t2 + t3 = 5 * √(2 S / a)

total distance travelled :  S + 2S + 3S =6 S

   => average speed = 3 / 5 * √(2 a S)

   maximum speed =  √(2 a S)

Ratio =  3/5