Q.For a projectile launched and landing at the same height, with air resistance neglected, the angle of projection that gives the maximum horizontal range is:

The range of a projectile launched at speed v and angle θ (measured from the horizontal) is given by

R = (v² sin(2θ))/g,

where g is the acceleration due to gravity and the launch and landing heights are equal.

To maximize R, we need to maximize sin(2θ). The sine function reaches its maximum value of 1 when its argument is 90°. Therefore,

2θ = 90° ⇒ θ = 45°.

At this angle, sin(2 × 45°) = sin(90°) = 1, and the range becomes

Rₘₐₓ = v²/g.

Why the other options are wrong:

• (a) 30°: At θ = 30°, sin(2 × 30°) = sin(60°) = √(3)/2 ≈ 0.866, which is less than 1. The range is only about 87% of the maximum.

• (c) 60°: At θ = 60°, sin(2 × 60°) = sin(120°) = √(3)/2 ≈ 0.866, the same as for 30°. Complementary angles (30° and 60°) give the same range, but neither is the maximum. …

Why this matters: the physics of projectile range

When you throw a ball, javelin, or shot put, the distance it travels depends on three things: how fast you release it, the angle at which you release it, and the height from which you release it. In the idealized case where the projectile lands at the same height it was launched—imagine throwing across a flat field—there is one angle that squeezes the maximum horizontal distance out of a given release speed.

The range of a projectile is given by

where v is the release speed, θ is the angle of projection above the horizontal, and g is gravitational acceleration.

Notice that the range depends on sin(2θ). The sine function reaches its maximum value of 1 when its argument is 90°. So sin(2θ) is maximized when 2θ = 90°, which means θ = 45°.

At this angle, the projectile spends equal time climbing and descending, and the horizontal and vertical components of velocity are balanced in a way that maximizes the product of flight time and horizontal speed. Launch too flat and the projectile hits the ground quickly; launch too steep and it spends time going up and down rather than forward. Forty-five degrees is the sweet spot.

Evaluating each option …