What is the simplified projectile motion equation for horizontal range in projectile motion without air resistance?

• A. Range = (v^2 × sin(2θ)) / g
• B. Range = (v × cosθ) / g
• C. Range = v^2 / (2g)
• D. Range = v × sinθ

Answer: (A)

In projectile motion without air resistance on level ground, the horizontal range is found by taking the horizontal speed and multiplying by the time the projectile stays in the air. The horizontal speed is v cos θ, and the time of flight for returning to the same height is T = 2 v sin θ / g. So the range is R = (v cos θ)(2 v sin θ / g) = (v^2 sin 2θ)/g, since sin 2θ = 2 sin θ cos θ.

This means the range depends on the square of the speed, the angle through sin 2θ, and gravity. The maximum range happens at θ = 45°, where sin 2θ = 1, giving R_max = v^2 / g. Remember, this result assumes launch and landing at the same height and no air resistance.

Other expressions don’t compute horizontal distance: one reflects vertical velocity or height terms, not the horizontal extent; the key is multiplying horizontal speed by the total time in the air and using the identity sin 2θ = 2 sin θ cos θ.