Question

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. 
Assertion (A): Range of a horizontal projectile is maximum when angle of projection is \(θ = 45°\).
Reason (R): Range is maximum when \(sin\ (2θ)=1\).
In the light of the above statements, choose the correct answer from the options given below

• A. (A) and (R) both are true and (R) is correct explanation of (A)
• B. (A) and (R) both are true but (R) is not correct explanation of (A)
• C. (A) is true and (R) is false
• D. Both (A) and (R) are false

Answer 1

The Correct Option is A

We know that:

Range \(R = \frac{\mu^2}{g}\,sin\,2\theta\)

R is maximum for \(2\theta=90^o\)

⇒ \(\theta=45^o\)

So, the correct option is (A): (A) and (R) both are true and (R) is correct explanation of (A).

Answer 2

Projectile Motion Range Problem 

Step 1: Horizontal Range Formula

For a ground-to-ground projectile, the horizontal range (R) is given by:

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

where \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

Step 2: Maximum Range Condition

For a given initial velocity (\( u \)), the range (R) is maximum when \( \sin 2\theta \) is maximum. The maximum value of \( \sin 2\theta \) is 1.

Step 3: Angle for Maximum Range

Since the maximum value of \( \sin 2\theta \) is 1, we have:

\( \sin 2\theta = 1 \)

\( 2\theta = \frac{\pi}{2} \)

\( \theta = \frac{\pi}{4} \text{ radians} = 45^\circ \)

Step 4: Analyze Assertion and Reason

Assertion A states that the range is maximum when the projection angle is 45°. This is correct, as shown in Step 3.

Reason R states that for maximum range, \( \sin 2\theta \) should be equal to one. This is also correct, as shown in Step 2.

Furthermore, Reason R explains Assertion A because the maximum value of \( \sin 2\theta \) (which is 1) occurs when \( \theta = 45^\circ \), leading to the maximum range.

Conclusion:

Both Assertion A and Reason R are correct, and Reason R is the correct explanation of Assertion A (Option 1).