Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Range is maximum at θ = 45º.
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. Both assertion (A) and reason (R) are true & reason (R) is the correct explanation of assertion.
• B. Both assertion (A) and reason (R) are true but reason (R is not correct explanation of assertion.
• C. Assertion (A) is true and reason (R) is false.
• D. Assertion (A) is false and reason (R) is false.

Answer 1

The Correct Option is A

The correct option is(A): Both assertion (A) and reason (R) are true & reason (R) is the correct explanation of assertion.

\(R=\frac{u^{2}sin(2\theta)}{g}\)
For the R_max
\(sin(2\theta)=1\)
\(sin(2\theta)=90^{o}\)
\(\theta=45^{o}\)

Answer 2

The assertion (A) is true. The range of a horizontal projectile is maximum when the angle of projection is \(45^{\circ}\). This can be shown using calculus or by using the fact that the horizontal and vertical components of the projectile's velocity are independent of each other.
The reason (R) is also true. The range of a projectile can be calculated using the formula:
\(R=(\frac{v^2}{g})\times sin(2\theta)\)
where v is the initial velocity of the projectile, g is the acceleration due to gravity, and θ is the angle of projection. To maximize the range, we need to find the value of θ that maximizes sin(2θ). Taking the derivative of sin(2θ) with respect to θ and setting it to zero, we get:
\(cos(2\theta) = 0\)
which simplifies to:
\(2\theta=(2n+1)\frac{\pi}{2}\)
where n is an integer. The value of θ that maximizes sin(2θ) is 
\(\theta=(2n+1)\frac{\pi}{4}\)
Since we are interested in the maximum range, we choose the value of θ that gives the largest value of sin(2θ), which is θ = \(45^{\circ}\)
Therefore, both (A) and (R) are true, and (R) is a correct explanation of (A). 
The correct answer is (A).