JEE Physics 2023 Question with Solution

Given: A projectile is projected with speed $u$ at angle $\theta$.

Find: Whether Assertion (A) and Reason (R) are true, and whether R correctly explains A.

For the range of a projectile,

$$R = \frac{u^2 \sin(2\theta)}{g}$$

For maximum range, the factor $\sin(2\theta)$ must be maximum.

The maximum value of $\sin(2\theta)$ is $1$. So,

$$\sin(2\theta) = 1$$

which gives

$$2\theta = 90^\circ$$

and therefore,

$$\theta = 45^\circ$$

Hence, the range is maximum when the angle of projection is $45^\circ$.

Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The correct option is C.

Given: The statement about maximum range of a projectile and the reason involving $\sin(2\theta)$.

Find: The correct assertion-reason option.

The assertion is true because the range of a projectile is given by

$$R = \left(\frac{v^2}{g}\right) \sin(2\theta)$$

To maximize $R$, we must maximize $\sin(2\theta)$. A maximum occurs when

$$\cos(2\theta) = 0$$

Thus,

$$2\theta = (2n+1)\frac{\pi}{2}$$

so

$$\theta = (2n+1)\frac{\pi}{4}$$

For the physically relevant acute angle giving maximum range,

$$\theta = 45^\circ$$

Thus, the reason is true and it correctly explains why the assertion is true.

Therefore, the correct option is C.