Question

A stone is thrown upwards with a speed u from the top of a tower. It reaches the ground with a velocity 3u. The height of the tower is
A. $\dfrac{{{{\text{u}}^2}}}{{\text{g}}}$
B. $\dfrac{{{\text{2}}{{\text{u}}^2}}}{{\text{g}}}$
C. $\dfrac{{{\text{3}}{{\text{u}}^2}}}{{\text{g}}}$
D. $\dfrac{{{\text{4}}{{\text{u}}^2}}}{{\text{g}}}$

Answer 1

- Hint: To solve this question, we will use the third equation of motion, which is ${{\text{v}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2as}}$. In this equation, we will put values of v and u to solve the given problem.

Complete step-by-step solution -

Now, let the height of the tower be h. The initial velocity(u) = u. The final velocity (v) = 3u. Now, we have to find the height of the tower. So, we will use the third equation of motion which is ${{\text{v}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2as}}$. The other two equations of motion are
(1) v = u + at
(2) ${\text{s = ut + }}\dfrac{1}{2}{\text{a}}{{\text{t}}^2}$
Where u is the initial velocity, v is the final velocity, t is the time taken and s is the distance covered.
Now, we are using the equation ${{\text{v}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2as}}$. As, we the height of tower is h, so the equation becomes
${{\text{v}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2ah}}$
Now, as the ball is thrown upward it will hit the ground travelling through the atmosphere, so there is no acceleration except g (known as gravitational acceleration or acceleration due to gravity of Earth). So, the equation becomes,
${{\text{v}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2gh}}$
So, applying the value of v and u in the above equation, we get
${{\text{(3u)}}^2}{\text{ = }}{{\text{u}}^2}{\text{ + 2gh}}$
$9{{\text{u}}^2}{\text{ - }}{{\text{u}}^2}{\text{ = 2gh}}$
Therefore, h = $\dfrac{{{\text{4}}{{\text{u}}^2}}}{{\text{g}}}$
So, the height of the tower = h = $\dfrac{{{\text{4}}{{\text{u}}^2}}}{{\text{g}}}$
So, option (D) is correct.

Note: When we come up with such types of questions, we will use the equations of motion to solve the given problem. The first equation shows a relation between u, v and t. The second equation shows a relation with u, t and s. The third equation shows a relation between u, v and s. We will use the equation according to the value asked in question. For example, in the above question, we have to find the height of the tower and there is no description about time in the question. So, we have used the third equation of motion to find the height of the tower.