Question

There are two projectiles thrown at angles \( \theta_1 \) & \( \theta_2 \) such that their ranges are same. Their speeds of projection are also same and time periods are 10 sec and 5 sec respectively. Find the range.

• A. \(250\ \text{m}\)
• B. \(300\ \text{m}\)
• C. \(650\ \text{m}\)
• D. \(100\ \text{m}\)

Answer 1

The Correct Option is A

Concept: For projectiles having the same speed but producing the same range, the angles of projection must be complementary. Important projectile formulas: \[ T = \frac{2u\sin\theta}{g} \] \[ R = \frac{u^2 \sin 2\theta}{g} \] Also, \[ \sin 2\theta = 2\sin\theta\cos\theta \] These relations help express range in terms of the time of flight expressions.
Step 1:Using the time of flight for the first projectile. \[ T_1=\frac{2u\sin\theta}{g}=10 \] \[ u\sin\theta=\frac{10g}{2}=5g \]
Step 2:Using the time of flight for the second projectile. Since the angles are complementary, the second projectile involves \( \cos\theta \): \[ T_2=\frac{2u\cos\theta}{g}=5 \] \[ u\cos\theta=\frac{5g}{2} \]
Step 3:Finding the range. \[ R=\frac{u^2\sin 2\theta}{g} \] \[ R=\frac{2(u\sin\theta)(u\cos\theta)}{g} \] Substitute the obtained values: \[ R=\frac{2(5g)\left(\frac{5g}{2}\right)}{g} \] \[ R=\frac{25g^2}{g}=25g \] Taking \( g=10\ \text{m/s}^2 \): \[ R=25\times10=250\ \text{m} \] \[ \boxed{R=250\ \text{m}} \]