Question

An elevator is moving up with \[2.5\,{\text{m}}{{\text{s}}^{ - 1}}\] . A bolt in the elevator ceiling \[3\,{\text{m}}\] above the elevator falls. How long (in seconds) does it take for the bolt to fall on the floor of the elevator?
(A) \[0.73\]
(B) \[0.71\]
(C) \[0.56\]
(D) \[1.06\]

Answer 1

Hint:First of all, we will choose an equation which relates speed, velocity, time and acceleration. We will substitute the required values and manipulate accordingly to find time. The bolt will feel weightlessness if the lift descends downwards. But in this case, it is moving up.

Complete step by step answer:
In the given question, we are supplied with the following data:
The initial speed of the elevator is \[2.5\,{\text{m}}{{\text{s}}^{ - 1}}\].The position of the bolt is \[3\,{\text{m}}\] above the floor of the elevator.We are asked to find out the time it takes to hit the floor of the elevator, after it falls off from the ceiling of the elevator.
To begin with, as we know the distance between the ceiling and the floor of the elevator, we can use one of the equations from the laws of motion. We also know the initial velocity of the bolt along with the acceleration due to gravity. We can find time simply by solving that equation.
We have a formula, which relates distance, initial velocity, time and acceleration due to gravity, which is given by:
\[S = ut + \dfrac{1}{2}g{t^2}\] …… (1)
Where,
\[S\] indicates the distance that the bolt covers from ceiling to the floor.
\[u\] indicates initial velocity of the bolt.
\[t\] indicates time taken by the bolt to hit the floor.
\[g\] indicates acceleration due to gravity.
Now, we substitute the required values in the equation (1) and we get:
$S = ut + \dfrac{1}{2}g{t^2} \\
\Rightarrow 3 = 2.5t + \dfrac{1}{2} \times 10{t^2} \\
\Rightarrow 6 = 5t + 10{t^2} \\
\Rightarrow 10{t^2} + 5t - 6 = 0 \\$
We have found out a quadratic equation, whose roots will give us the required answer.
We apply the formula which finds the roots as:
$t = \dfrac{{ - 5 \pm \sqrt {{5^2} - 4 \times 10 \times \left( { - 6} \right)} }}{{2 \times 10}} \\
\Rightarrow t = \dfrac{{ - 5 \pm \sqrt {25 + 240} }}{{20}} \\
\Rightarrow t = \dfrac{{ - 5 \pm \sqrt {265} }}{{20}} \\
\Rightarrow t = \dfrac{{ - 5 \pm 16.27}}{{20}} \\$
We know, time is always positive so we take the positive value from it i.e.
$t = \dfrac{{ - 5 + 16.27}}{{20}} \\
\Rightarrow t = \dfrac{{11.27}}{{20}} \\
\therefore t = 0.56\,{\text{s}}$
Hence, the time it takes to hit the floor of the elevator, after it falls off from the ceiling of the elevator is \[0.56\,{\text{s}}\] .

The correct option is C.

Note:While solving this problem, many students seem to have a confusion regarding finding time. A quadratic equation has two roots, but one of them can also be negative. However, time can’t be negative so we should reject such values. The bolt is also moving up, so it takes less time to hit the floor. It would have taken some time long if the lift was descending.