Question

A lift of height 3.84m starts from rest and accelerated up at the rate of 2m/s2,1.5s later a bolt from its roof is distached from it time taken by the bolt to strike the flour of lift,distance and displacement with respect ground in that time

Answer 1

Let g = 10 m/s²
height of lift = h = 3.84 m
acceleration = a = 2 m/s²
velocity of the ceiling , floor walls of the lift after 1.5 sec, from the starting from rest
     = v1 = a t = 3 m/sec
At this point of time the bolt has travelled upwards:
        s1 = 1/2 * a * t² = 1/2 * 2 * 1.5² = 2.25 meters
  So it is at  s1 + h = 6.09 meters  above the ground.
At this point of time the floor is at s1 = 2.25 meters from the ground.

When the bolt is detached at t = 1.5sec, it continues to move upwards.  Its velocity is 3 m/sec upwards.  It decelerates due to acceleration due to gravity acting downwards.  Its distance from the floor is equal to h. 

let us say that they meet at 1.5 + T sec. after the starting of the lift at t = 0 sec.
In the time duration T sec, the floor travels upwards = d2.
   d2 = v1 * T + 1/2 * a * T² = 3 T + T² meters.

In the time duration T sec., the displacement of bolt downwards is
       d1 = - v1 * T + 1/2 * g * T²
               = -3 * T + 5 T²

we have that   d2 + d1 = h = height of lift
                    3 T + T² - 3 T + 5 T² = h
                     6 T² - 3.84 = 0
                     T =  0.80 seconds.

At t = 1.50 + 0.80 sec = 2.30 sec, the floor will be at a height from the ground:
           = 0 * 2.30 sec + 1/2 * 2 m/s² * 2.30² = 5.29 meters.  (displacement from ground)
Clearly, the bolt is also at the same height from the ground. Same displacement.

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The floor of the lift travels the same distance as the displacement.

But the bolt travels more distance than the displacement.  Bolt travels upwards with an initial speed of 3.0 m/s upwards from time t = 1.50 sec. and then it will reach a top point and then fall downwards.
     the time duration it takes to reach its top = (3.0 m/s) / 10 m/s² = 0.30 sec.
   In this time duration, it travels upwards :  avg velocity * time duration
                    = (3.0 m/s + 0)/2 * 0.30 sec = 0.45 meters
   Then it will travel also 0.45 meters downward, to reach the point where it got detached from the ceiling of lift.  Hence, the bolt travels 0.90 meters, even though its displacement is 0 from that at  time t = 1.50 sec.

Hence, the bolt travels totally a distance of 5.29 meters + 0.90 meters = 6.19 meters.