Question

A body is thrown up in a lift with a velocity $u$ relative to the lift and the time of flight is $t$ . Show that the lift is moving up with an acceleration $\dfrac{{2u - tg}}{t}$.

Answer 1

In order to answer the question to show that the lift is moving up with the given acceleration we will use the concept of relative motion equation. In this question we will use the second equation of Newton to find the value acceleration of lift. Acceleration due to gravity and lift will be added because both are in opposite directions.

Formula used:
$S = ut + \dfrac{1}{2}a{t^2}$
Where, $s =$ Displacement, $u =$ Initial velocity, $a =$ acceleration and $t =$ time $t$.

Complete step by step answer:
Now, let us come to the given question; let us consider that the acceleration of the lift be $'a'$ in the upward direction. And, acceleration of the body with respect to the ground $= g$ (downward). Thus, the acceleration of the body with respect to the lift will be;
${a_b} = g + a$
So, in the lift frame: $S = ut + \dfrac{1}{2}{a_b}{t^2}$
Where we know that $'t'$ is the time of the flight
$0 = ut - \dfrac{{\left( {g + a} \right){t^2}}}{2} \\\ \Rightarrow a = \dfrac{{2u - tg}}{2} \\\$

Hence, it is proved that the lift is moving up with an acceleration $\dfrac{{2u - tg}}{t}$.

Note: The second equation of motion is $s{\text{ }} = {\text{ }}ut + \dfrac{1}{2}{\text{ }}a{t^2}$ . It's also known as the position-time relationship. When the numbers for time, acceleration, and beginning velocity are known, we can use this formula to compute the displacement. $s$ stands for displacement. Because this equation includes four values, the fourth value may be determined if the first three are known.