Question

A lift is accelerating downwards at the rate of 4$m/{s^2}$. At an instant when the velocity of the lift is 8 m/s, a bolt starts to drop from the ceiling of the lift. Find the time taken for the bolt to reach the floor of the lift, if the ceiling is 4 m above the floor.
A. 1.15s
B. 2.01s
C. 3.2s
D. 4s

Answer 1

Hint: The lift and the bolt are accelerating in the same direction so the relative acceleration between the bot and the lift will play a role in the motion of the bolt. We can make use of the equations of motion in order to study the motion of the given bolt as it falls in the lift in order to obtain the required answer.
Formula used:
Equations of motion for a mechanical system are given as follows:
$
  v = u + at \\
  S = ut + \dfrac{1}{2}a{t^2} \\
  {v^2} - {u^2} = 2aS \\
 $

Complete answer:
We are given a lift which is accelerating in the downward direction with an acceleration whose value is given as
$a = 4m/{s^2}$
Now a bolt starts to drop from the ceiling of the lift when the velocity of the lift is
$v = 8m/s$
As the lift accelerates, the velocity of the lift will change. The length of the lift is given as
$S = 4m$
The bolt will accelerate with acceleration equal to the acceleration due to gravity as it falls down the roof of the lift. This acceleration is given as
$g = 10m/{s^2}$
As the bolt and the lift are falling in the same direction, so the relative acceleration of the bolt and the lift can be written as
$a' = g - a = 10 - 4 = 6m/{s^2}$
The initial velocity of the bolt is zero inside the lift.
$u = 0$
So, we can make use of the second equation of motion in the following way.
$S = ut + \dfrac{1}{2}a'{t^2}$
Now on inserting the known values, we get
$
  4 = 0 + \dfrac{1}{2} \times 6 \times {t^2} \\
  \Rightarrow 4 = 3{t^2} \\
  \Rightarrow t = \dfrac{2}{{\sqrt 3 }} = 1.15s \\
 $

So, the correct answer is “Option A”.

Note:
It should be noted that the lift is not undergoing the free fall under gravity. If the lift falls freely under gravity, the acceleration of lift will be equal to the acceleration of the bolt. As a result, their relative acceleration will be zero and the bolt will never reach the floor of the lift.