Question 21: Motion in 1D

Question

A body starts moving from rest with constant acceleration covers displacement $$S_1$$ in first $$(p-1)$$ seconds and $$\mathrm{S}_2$$ in first $$p$$ seconds. The displacement $$\mathrm{S}_1+\mathrm{S}_2$$ will be made in time :

Accepted Answer

Let's denote the constant acceleration with which the body moves as $$a$$. We know that the displacement $$S$$ covered by a body starting from rest under a constant acceleration $$a$$ in time $$t$$ is given by the equation of motion: $$S = \frac{1}{2} a t^2$$.
Considering the displacement $$\mathrm{S}_1$$ in first $$(p-1)$$ seconds, we apply the equation of motion:
$$ \mathrm{S}_1 = \frac{1}{2} a (p-1)^2 $$
Similarly, for the displacement $$\mathrm{S}_2$$ in first $$p$$ seconds:
$$ \mathrm{S}_2 = \frac{1}{2} a p^2 $$
To find out the total time it will take to cover the displacement $$\mathrm{S}_1+\mathrm{S}_2$$, we first find the sum of these two displacements:
$$ \mathrm{S}_1+\mathrm{S}_2 = \frac{1}{2} a (p-1)^2 + \frac{1}{2} a p^2 $$
Let's simplify this:
$$ \mathrm{S}_1+\mathrm{S}_2 = \frac{1}{2} a \left((p-1)^2 + p^2ight) $$
$$ \mathrm{S}_1+\mathrm{S}_2 = \frac{1}{2} a \left(p^2 - 2p + 1 + p^2ight) $$
$$ \mathrm{S}_1+\mathrm{S}_2 = \frac{1}{2} a \left(2p^2 - 2p + 1ight) $$
$$ \mathrm{S}_1+\mathrm{S}_2 = a \left(\frac{2p^2 - 2p + 1}{2}ight) $$
If we consider the total displacement $$\mathrm{S}_1+\mathrm{S}_2$$ is to be covered in a time $$t$$ seconds from rest, we should set this equal to the equation of motion:
$$ \mathrm{S}_1+\mathrm{S}_2 = \frac{1}{2} a t^2 $$
Equating the two equations:
$$ a \left(\frac{2p^2 - 2p + 1}{2}ight) = \frac{1}{2} a t^2 $$
Since $$a 
eq 0$$, we can simplify by dividing both sides by $$ \frac{1}{2} a$$:
$$ \frac{2p^2 - 2p + 1}{2} = \frac{t^2}{2} $$
$$ 2p^2 - 2p + 1 = t^2 $$
To find $$t$$, we take the square root of both sides:
$$ t = \sqrt{2p^2 - 2p + 1} $$
Therefore, the time taken to cover the displacement $$\mathrm{S}_1+\mathrm{S}_2$$ will be:
$$ t = \sqrt{(2p^2 - 2p + 1)}\ s $$
Hence, the correct option would be:
Option D: $$ \sqrt{(2p^2 - 2p + 1)}\ s $$

Answer

$$(2 p+1) s$$

Answer

$$(2 p-1) s$$

Answer

$$\left(2 p^2-2 p+1\right) s$$

Answer

$$\sqrt{\left(2 p^2-2 p+1\right)} s$$

Motion in 1D

A body starts moving from rest with constant acceleration covers displacement $S_1 in first (p-1) seconds and \mathrm{S}_2 in first p seconds. The displacement \mathrm{S}_1+\mathrm{S}_2$ will be made in time :

Choose the correct answer: