Question

This question consists of two statements, each printed as an assertion and a reason. While answering this question, you are required to choose any one of the following five responses.
Assertion: If a body is thrown upwards, the distance covered by it in the last second of upward motion is about 5m irrespective of its initial speed.
Reason: The distance covered in the last second of upward motion is equal to that covered in the first second of downward motion when the particle is dropped.
A) Both assertion and reason are true and the reason is the correct explanation of assertion
B) Both assertion and reason are true but the reason is not the correct explanation of the assertion
C) Assertion is true but reason is false.
D) Both assertion and reason are false.
E) Reason is true but assertion is false.

Answer 1

Hint
Use Newton’s second equation of Motion $\left( {s = ut + \dfrac{1}{2}a{t^2}} \right)$ to compute the distance travelled by the body in the last second of upward motion and first second of downward motion to evaluate the statements. Given reason is true, determine whether assertion is true to check whether reason is the correct explanation of the assertion.

Complete answer:
Firstly, we compute the distance travelled by a body thrown upwards, let its initial velocity be u and the time taken to reach maximum height be t. Then, we have, from Newton’s second equation of motion,
$ \Rightarrow s = ut + \dfrac{1}{2}a{t^2}$
The distance that the body travels in time t, experiencing acceleration due to gravity g is therefore,
$ \Rightarrow {s_t} = ut - \dfrac{1}{2}g{t^2}$ (Since g acts opposite to the displacement of the object)
Now, at time t-1 seconds, the distance covered by the body is given by,
$ \Rightarrow {s_{t - 1}} = u(t - 1) - \dfrac{1}{2}g{(t - 1)^2}$
Simplifying the terms we get,
$ \Rightarrow {s_{t - 1}} = ut - u - \dfrac{1}{2}g({t^2} + 1 - 2t) = ut - u - \dfrac{1}{2}g{t^2} - \dfrac{1}{2}g + gt$
Rearranging the terms in the above, we get,
$ \Rightarrow {s_{t - 1}} = ut - \dfrac{1}{2}g{t^2} - \dfrac{1}{2}g - (u - gt)$
From Newton’s first equation of motion, we know that,
$ \Rightarrow v = u - gt = 0$ as the object has zero velocity at time t at maximum height.
Using this in the equation for ${s_{t - 1}}$ we get,
$ \Rightarrow {s_{t - 1}} = ut - \dfrac{1}{2}g{t^2} - \dfrac{1}{2}g = ut - \dfrac{1}{2}g{t^2} - \dfrac{g}{2}$
Distance travelled by the body in the last second,
$ \Rightarrow d = {s_t} - {s_{t - 1}}$
Putting the value of ${s_t},{s_{t - 1}}$ as obtained, we get,
$ \Rightarrow d = ut - \dfrac{1}{2}g{t^2} - \left( {ut - \dfrac{1}{2}g{t^2} - \dfrac{g}{2}} \right)$
Opening the brackets we get,
$ \Rightarrow d = ut - \dfrac{1}{2}g{t^2} - ut + \dfrac{1}{2}g{t^2} + \dfrac{g}{2}$
Cancelling the terms, we get
$ \Rightarrow d = \dfrac{g}{2}$
Since, $g \approx 10m/{s^2}$ , therefore, $d \approx 5m$ , which means that if a body is thrown upwards, the distance covered by it in the last second of upward motion is about 5m irrespective of its initial speed. Hence, the assertion is true.
Now, to evaluate the reason, we compute the distance covered by a particle in the first second of downward motion, ${d_d}$
Using Newton’s second law of motion, we have,
$ \Rightarrow s = ut + \dfrac{1}{2}a{t^2}$
Since the object is dropped, we have the initial velocity, u=0, acceleration = g, and distance travelled in the first second $(t = 1)$ of downward motion=${d_d}$ , we have,
$ \Rightarrow {d_d} = 0 + \dfrac{1}{2}g{(1)^2} = \dfrac{g}{2} \approx 5m$
Hence, $d = {d_d}$ ; therefore the reason is also correct.
Now, both the assertion and reason statements are correct, also we know that assuming the reason to be correct, we can prove the assertion statement, i.e. : if the distance covered in the last second of upward motion is equal to that covered in the first second of downward motion when the particle is dropped, we know that the particle will travel roughly 5m in the last second irrespective of its initial speed as the distance covered by the particle in the first second of downward motion clearly is independent of the initial velocity of the object.
Hence, the reason is the correct explanation of the assertion.
Option (A) is the correct answer.

Note
The general methodology for solving such questions is to evaluate the assessment and reasoning statements for their truth first. If either or both are wrong then we can mark the correct option accordingly, if both are correct, there is an extra step to determine whether the reason is the correct explanation of assertion or not.