Question

A stone is projected vertically upwards with speed 'v'. Another stone of same mass is projected at an angle of $60^\circ$ with the vertical with the same speed 'v'. The ratio of their potential energies at the highest points of their journey is [$\sin 30^\circ = \cos 60^\circ = 0.5$, $\cos 30^\circ = \sin 60^\circ = \frac{\sqrt{3{2}$]

• A. 4:1
• B. 2:1
• C. 3:2
• D. 1:1

Hint:

Logic Tip: The potential energy at the highest point is determined exclusively by the vertical component of the launch velocity ($v_y = v \sin\theta$). Since $PE \propto v_y^2$, the ratio is simply $\frac{(v \sin 90^\circ)^2}{(v \sin 30^\circ)^2} = \frac{1^2}{0.5^2} = \frac{1}{0.25} = 4$.

Answer 1

The Correct Option is A

Concept:
The maximum height reached by a projectile launched with velocity $u$ at an angle $\theta$ with the horizontal is $H_{max} = \frac{u^2 \sin^2 \theta}{2g}$. The potential energy at this maximum height is $PE = mgH_{max}$.
Step 1: Calculate the maximum potential energy of the first stone.
The first stone is projected vertically upwards. This corresponds to an angle of $90^\circ$ with the horizontal. $$H_1 = \frac{v^2 \sin^2(90^\circ)}{2g} = \frac{v^2 (1)^2}{2g} = \frac{v^2}{2g}$$ The potential energy at the highest point is: $$PE_1 = mgH_1 = mg\left(\frac{v^2}{2g}\right) = \frac{1}{2}mv^2$$
Step 2: Calculate the maximum potential energy of the second stone.
The second stone is projected at an angle of $60^\circ$ with the \textit{vertical}. This means the angle of projection with the \textit{horizontal} is $\theta = 90^\circ - 60^\circ = 30^\circ$. $$H_2 = \frac{v^2 \sin^2(30^\circ)}{2g} = \frac{v^2 \left(\frac{1}{2}\right)^2}{2g} = \frac{v^2 \left(\frac{1}{4}\right)}{2g} = \frac{v^2}{8g}$$ The potential energy at the highest point is: $$PE_2 = mgH_2 = mg\left(\frac{v^2}{8g}\right) = \frac{1}{8}mv^2$$
Step 3: Calculate the ratio of the potential energies.
$$\text{Ratio} = \frac{PE_1}{PE_2} = \frac{\frac{1}{2}mv^2}{\frac{1}{8}mv^2}$$ $$\text{Ratio} = \frac{\frac{1}{2{\frac{1}{8 = \frac{8}{2} = \frac{4}{1}$$ The ratio is 4:1.