Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are $5$ s and $10$ s, respectively, then the value of $R$ is ______ m. (Take $g = 10\ \text{m/s}^2$)
Answer: (A) $250$
Concept. At the same speed, $R=\dfrac{u^2\sin2\theta}{g}$ takes the same value for $\theta$ and $90^\circ-\theta$. The two flight times are then $\dfrac{2u\sin\theta}{g}$ and $\dfrac{2u\cos\theta}{g}$, and their product is directly related to $R$.
Step 1. Write the two flight times.
$$T_1=\frac{2u\sin\theta}{g},\qquad T_2=\frac{2u\sin(90^\circ-\theta)}{g}=\frac{2u\cos\theta}{g}$$
Step 2. Multiply them.
$$T_1T_2=\frac{4u^2\sin\theta\cos\theta}{g^2}=\frac2g\cdot\frac{u^2\sin2\theta}{g}=\frac{2R}{g}$$
Step 3. Substitute.
$$R=\frac{gT_1T_2}{2}=\frac{10\times5\times10}{2}=250\ \text{m}$$
Answer. $250$ m (Option A).
Solution by Sreeraj P, M.Sc Physics
