Physics · Motion In A Plane · NEET
Range = horizontal velocity × time of flight. The horizontal velocity is u cos θ (it never changes because gravity acts only downward). The time of flight is T = 2u sin θ / g. Multiply them: R = (u cos θ)(2u sin θ / g) = 2u^2 sin θ cos θ / g. Using the identity 2 sin θ cos θ = sin(2θ), this becomes R = u^2 sin(2θ) / g. The sin(2θ) appears only after combining the two directions, so range naturally carries the double angle.
Yes. Range depends on sin(2θ). For 30°, sin(2×30°) = sin 60°. For 60°, sin(2×60°) = sin 120° = sin 60°. Both give the same value, so the ranges are equal. In general, angles θ and (90° − θ) always give the same range because sin(2θ) = sin(180° − 2θ). The 60° throw goes higher and stays in the air longer, but the slower horizontal speed exactly cancels this, so both land at the same distance.
No. In ideal projectile motion (no air resistance), there is no horizontal force, so the horizontal component u cos θ stays constant the whole time. This is why range is simply (constant horizontal speed) × (time of flight). Only the vertical velocity changes due to gravity g. This split — constant horizontal, changing vertical — is the core idea NEET tests again and again.
It applies only when the projectile lands back at the SAME height from which it was launched (flat ground, symmetric path). If it is thrown from a cliff and lands lower, or thrown up onto a platform, this formula fails — you must use x = (u cos θ) × t with the actual time of flight found from the vertical equation. NTA often hides this: a projectile from a height is NOT a same-level range problem.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
R = u^2 sin(2θ) / g, where u is the launch speed, θ is the angle of projection above the horizontal, and g is the acceleration due to gravity (about 9.8 m/s^2). It is valid only when the projectile lands at the same height it started.
At θ = 45°, because sin(2θ) = sin 90° = 1, its largest value. The maximum range is R(max) = u^2 / g. This is covered in detail on the next concept page, angle for maximum range.
Because sin(2×15°) = sin 30° and sin(2×75°) = sin 150° = sin 30°. Any pair θ and (90° − θ) gives equal ranges since their double angles are supplementary and have the same sine.
No. The range formula R = u^2 sin(2θ) / g has no mass term. Without air resistance, all projectiles launched with the same speed and angle have the same range, whatever their mass.
Range = horizontal velocity × time of flight = (u cos θ) × (2u sin θ / g). Since horizontal speed is constant, the range is directly set by how long the projectile stays in the air.
