Question

A metal ball of mass m is projected at an angle $\theta$ with the horizontal with an initial velocity $ u $. If the mass and angle of projection are doubled keeping the initial velocity the same, the ratio of the maximum height attained in the former to the latter case is:

• A. 1 : 2
• B. 2 : 1
• C. 1 : 3
• D. 3 : 1

Hint:

When solving problems related to the maximum height of a projectile, use the formula \( H = \frac{u^2 \sin^2(\theta)}{2g} \). Remember, changing the angle of projection affects the sine term, and doubling the angle leads to a change in the sine squared term.

Answer 1

The Correct Option is C

The maximum height \( H \) attained by a projectile is given by the formula: \[ H = \frac{u^2 \sin^2(\theta)}{2g}, \] where: - \( u \) is the initial velocity, - \( \theta \) is the angle of projection, - \( g \) is the acceleration due to gravity. 
Case 1: Original values In the first case, let the initial velocity be \( u \) and the angle of projection be \( \theta \). The maximum height attained in the first case is: \[ H_1 = \frac{u^2 \sin^2(\theta)}{2g}. \] 
Case 2: Doubled mass and angle In the second case, the mass is doubled, but the initial velocity remains the same. The angle of projection is also doubled, so the angle becomes \( 2\theta \). The maximum height in the second case is: \[ H_2 = \frac{u^2 \sin^2(2\theta)}{2g}. \] Using the double angle identity for sine, we know that: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta). \] 
Thus, the maximum height in the second case becomes: \[ H_2 = \frac{u^2 \left(2 \sin(\theta) \cos(\theta)\right)^2}{2g} = \frac{4u^2 \sin^2(\theta) \cos^2(\theta)}{2g}. \] 
Now, the ratio of the maximum heights in the first case to the second case is: \[ \frac{H_1}{H_2} = \frac{\frac{u^2 \sin^2(\theta)}{2g}}{\frac{4u^2 \sin^2(\theta) \cos^2(\theta)}{2g}} = \frac{1}{4 \cos^2(\theta)}. \] For \( \theta = 30^\circ \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so: \[ \frac{H_1}{H_2} = \frac{1}{4 \left(\frac{\sqrt{3}}{2}\right)^2} = \frac{1}{4 \times \frac{3}{4}} = \frac{1}{3}. \] 
Thus, the ratio of the maximum heights attained is \( 1 : 3 \), and the correct answer is option (C).