Question

If a coin is dropped in a lift it takes \[{t_1}\]​ time to reach the floor and takes \[{t_2}\] time when lift is moving up with constant acceleration, when which one of the following relations is correct?

(A) \[{t_1} = {t_2}\]
(B) \[{t_1} > {t_2}\]
(C) \[{t_2} > {t_1}\]
(D) \[{t_1} > > {t_2}\]

Answer 1

Hint It is given that a coin dropped inside a lift takes time \[{t_1}\] to reach the floor. In this case acceleration is zero. The , when the lift moves up in a constant acceleration , the time of fall is now \[{t_2}\]. Use motion equations, substantiate a relationship between the time periods.

Complete Step By Step Solution
Let us divide our problem into two scenarios. In the first scenario it is given that the lift is stationary and doesn’t undergo any motion as such. Inside the lift, there is a coin dropped. Now, the acceleration of the coin is now said to be acceleration due to gravity and the height will be that of the distance it is dropped from. Let us assume that distance to be \[h\]. Now using second motion equation we get,
\[h = ut + \dfrac{1}{2}g{t^2}\], where h is vertical displacement, g is acceleration due to gravity and u is initial velocity of the object.
In our scenario,
\[ \Rightarrow h = (0){t_1} + \dfrac{1}{2}g{t_1}^2\]
Now in our second scenario, the lift moves upwards with a constant acceleration ‘a’. Now, at any given instance, the lift will rise by a certain height ‘s’. Now, the coin takes a time period \[{t_2}\]to reach the floor. The acceleration of the coin going down will be the sum of constant acceleration of the frame(lift) and acceleration due to gravity.
Now , since it is said that the lift will rise height ‘s’ at any given instance, the distance of fall will be the sum of original distance ‘h’ and lift rise ‘s’. Since lift is moving upwards, the distance will reduce by quite a great margin. Applying second equation of motion, we get
\[ \Rightarrow (h - s) = (0){t_2} + \dfrac{1}{2}(g + a){t_2}^2\]
Now from the above statements, we can conclude that time taken in the second scenario will be lesser than the first.

Hence, Option(b) is the right answer for the given question.

Note The major concept we use to solve this question is often referred to as a Non-inertial reference frame. A non-inertial reference frame is defined as a reference frame that undergoes acceleration in terms of an inertial frame.