Why is 45° the best angle?

On level ground with no air resistance the range is v² sin(2θ)/g, which is largest when 2θ = 90°. Launching from a height, or turning drag on, moves the best angle below 45°.

How do you work out the range of a projectile?

Split the launch velocity into v cos θ across and v sin θ up. The time of flight on level ground is 2v sin θ/g, and the range is the horizontal speed times that time.

What does air resistance change?

Drag opposes the velocity, so the ball slows down sideways and the descent is steeper than the climb. Range and peak height both drop, and the path is no longer a parabola.

What angle do I launch at to hit something a set distance away?

On level ground with no air, sin 2θ = gR/v², which has two answers. At 20 m/s, to land 30 m away: sin 2θ = 9.81 × 30 / 400 = 0.736, so θ = 23.7° or 66.3°, a flat shot or a lob. If gR/v² is above 1 it is out of reach. Pick Angles for a range under Solve for to see each step, from a height too.

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Projectile motion

Launch angle, speed and gravity decide the arc. Compare runs with and without air resistance.

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Height vs time

Readouts

What's happening

Horizontal and vertical motion are independent. Gravity only pulls the vertical velocity down, so the path is a parabola. With air resistance on, a drag force opposing the velocity shortens the range and makes the arc asymmetric. Your last three arcs stay on screen so you can compare settings directly.

x = v cos(θ) ty = v sin(θ) t − ½gt²

A-Level Physics (AQA 3.4.1.5, OCR, Edexcel) and A-Level Maths mechanics: projectiles. GCSE Physics: independence of horizontal and vertical motion.

Challenge

Predict first: which angle gives the longest range at fixed speed? Now turn air resistance on: does the best angle stay at 45°, or move lower? Launch and check.

FAQ

Why is 45° the best angle?: On level ground with no air resistance the range is v² sin(2θ)/g, which is largest when 2θ = 90°. Launching from a height, or turning drag on, moves the best angle below 45°.
How do you work out the range of a projectile?: Split the launch velocity into v cos θ across and v sin θ up. The time of flight on level ground is 2v sin θ/g, and the range is the horizontal speed times that time.
What does air resistance change?: Drag opposes the velocity, so the ball slows down sideways and the descent is steeper than the climb. Range and peak height both drop, and the path is no longer a parabola.
What angle do I launch at to hit something a set distance away?: On level ground with no air, sin 2θ = gR/v², which has two answers. At 20 m/s, to land 30 m away: sin 2θ = 9.81 × 30 / 400 = 0.736, so θ = 23.7° or 66.3°, a flat shot or a lob. If gR/v² is above 1 it is out of reach. Pick Angles for a range under Solve for to see each step, from a height too.