Question

A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance $R$ from it. If $t_1$ and $t_2$ are the values of the time taken by it to hit the target in two possible ways, the product $t_1t_2$ is

• A. $\dfrac{R}{g}$
• B. $\dfrac{2R}{g}$
• C. $\dfrac{R}{2g}$
• D. $\dfrac{R}{4g}$

Hint:

For a given range with the same speed, the two angles of projection are complementary, and their times of flight can be related using projectile motion formulas.

Answer 1

The Correct Option is B

Step 1: For a given range $R$ with the same initial speed $u$, a projectile can be fired at two complementary angles $\theta$ and $(90^\circ-\theta)$.
Step 2: Time of flight for a projectile is: \[ t=\frac{2u\sin\theta}{g} \] Hence, \[ t_1=\frac{2u\sin\theta}{g}, \qquad t_2=\frac{2u\cos\theta}{g} \]
Step 3: Find the product $t_1t_2$: \[ t_1t_2=\frac{4u^2\sin\theta\cos\theta}{g^2} =\frac{2u^2\sin2\theta}{g^2} \]
Step 4: The range of a projectile is: \[ R=\frac{u^2\sin2\theta}{g} \]
Step 5: Substitute $u^2\sin2\theta = Rg$: \[ t_1t_2=\frac{2Rg}{g^2}=\frac{2R}{g} \]