Question

A particle is projected with velocity \( u \) so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as \( \frac{nu^2}{25g} \), where value of \( n \) is : (Given 'g' is the acceleration due to gravity).

• A. 6
• B. 18
• C. 12
• D. 24

Hint:

Use the formulas for the horizontal range and maximum height of a projectile. Set up the given condition relating the range and maximum height to find the angle of projection. Once the angle is known (or trigonometric ratios of the angle are known), substitute these values back into the formula for the range to express it in the required form and find the value of \( n \).

Answer 1

The Correct Option is D

The objective is to determine the value of \( n \) where the projectile's horizontal range \( R \) is thrice its maximum height \( H \). The given expression for the range is \( \frac{nu^2}{25g} \).

1. The formulas for the horizontal range \( R \) and maximum height \( H \) of a projectile are:
   • Horizontal range: \( R = \frac{u^2 \sin(2\theta)}{g} \)
   • Maximum height: \( H = \frac{u^2 \sin^2(\theta)}{2g} \)
2. The problem states that \( R = 3H \).
3. Substituting the expressions for \( R \) and \( H \) yields: \( \frac{u^2 \sin(2\theta)}{g} = 3 \times \frac{u^2 \sin^2(\theta)}{2g} \).
4. Simplifying the equation gives: \( \sin(2\theta) = \frac{3}{2} \sin^2(\theta) \).
5. Using the identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \), we get: \( 2\sin(\theta)\cos(\theta) = \frac{3}{2} \sin^2(\theta) \).
6. After canceling terms, we have: \( 4\cos(\theta) = 3\sin(\theta) \).
7. This implies: \( \tan(\theta) = \frac{4}{3} \).
8. From this, we find:
   • \( \sin(\theta) = \frac{4}{5} \)
   • \( \cos(\theta) = \frac{3}{5} \)
9. Substituting these values back to find the range: \( R = \frac{u^2 \times \frac{3}{5} \times \frac{4}{5}}{g} = \frac{12u^2}{25g} \).
10. Equating this to the given range expression \( \frac{nu^2}{25g} \) to solve for \( n \): \( \frac{nu^2}{25g} = \frac{12u^2}{25g} \).
11. Solving for \( n \): \( n = 12 \times 2 = 24 \).
12. The value of \( n \) is determined to be \( 24 \).

Consequently, the correct option is 24.