The Best Angle for Maximum Range (It Isn't Always 45°)

If you launch something from ground level and there's no air to slow it down, the best angle for maximum range is 45°. Aim lower and it hits the ground too soon. Aim higher and it wastes its speed climbing instead of travelling. That's the textbook answer, and it's right as far as it goes. But real throws rarely start at ground level and real air is never absent, so the angle that actually wins is usually a little lower. Here's the reasoning, with numbers you can check yourself.

Speed has to do two jobs at once

When a ball leaves your hand at 20 m/s, that speed is shared between two directions. The horizontal part carries it forward. The vertical part fights gravity and decides how long the ball stays up.

You can't give both parts full strength. Fire it flat (0°) and all 20 m/s goes forward, but the ball is in the air for no time at all, so it goes nowhere. Fire it straight up (90°) and it stays up for ages, then lands on your feet. Somewhere between those extremes is a sweet spot where "time in the air" multiplied by "forward speed" is as big as it can be.

The projectile range formula, explained

On level ground with no air resistance, it takes three short steps.

1. The forward speed is v·cos θ, and it never changes, because nothing pushes the ball sideways once it's flying.
2. The time in the air is t = 2v·sin θ ÷ g. The ball rises until gravity has cancelled its upward speed, then takes the same time to fall back.
3. Range is forward speed multiplied by time:

R = v·cos θ × 2v·sin θ ÷ g = v²·sin 2θ ÷ g

The last step uses the identity 2·sin θ·cos θ = sin 2θ, which you may already have met in maths.

Why is 45 degrees the best launch angle?

Look at the formula again. For a given throw, v and g don't change, so the range depends only on sin 2θ. The largest value a sine can ever reach is 1, and sin 2θ = 1 when 2θ = 90°. So θ = 45°.

Here are real numbers for a 20 m/s launch on Earth (g = 9.81 m/s²):

Launch angle	Range	Time in the air	Highest point
15°	20.4 m	1.06 s	1.4 m
30°	35.3 m	2.04 s	5.1 m
45°	40.8 m	2.88 s	10.2 m
60°	35.3 m	3.53 s	15.3 m
75°	20.4 m	3.94 s	19.0 m

Complementary angles give the same range

Look at the pairs in that table. 30° and 60° both land at 35.3 m. 15° and 75° both land at 20.4 m. That isn't luck. Angles that add up to 90° are called complementary, and sin 2θ comes out identical for both of them (sin 60° is the same as sin 120°). The steep shot spends longer in the air but crawls forward. The flat shot moves forward fast but doesn't stay up long. The two effects cancel exactly.

This is genuinely useful. If you need to clear a wall, take the high angle. If you want the ball to arrive quickly, like a fast pass, take the low one. Same landing spot, very different journeys.

When 45° stops being the answer

The 45° rule rests on two assumptions: you launch from the same height you land at, and there's no air. Break either one and the best angle moves.

Launching from a height. A shot-putter lets go of the shot about two metres above the ground. Because the shot already has a head start on its fall, it can afford a flatter path, and the ideal angle drops below 45°. There's a human reason as well: people can push harder at lower angles. In competition the shot usually leaves the hand somewhere in the high 30s of degrees.

Throwing downhill or off a cliff works the same way. The further below your launch point you land, the flatter the best angle becomes.

Does air resistance change the best angle?

Yes, and it pulls it lower. Drag slows the ball for the whole flight, and a high, looping shot spends the longest in the air, so it loses the most. A flatter shot reaches the ground before drag can steal as much of its speed.

How much lower depends on what you throw. A heavy cannonball at a gentle speed barely notices the air, so its best angle stays close to 45°. A beach ball is light and big, so drag takes over. It slows quickly, then drops almost straight down, and its range shrinks dramatically.

Golf is its own special case. A drive leaves the club at only around 10° to 15°. Drag is part of the reason, but the bigger one is backspin: it gives the ball lift, a bit like a wing, and keeps it in the air far longer than gravity alone would allow.

What about on the Moon?

Take away the air and weaken gravity, and the 45° rule comes back in its purest form. With g = 1.62 m/s² instead of 9.81, the same 20 m/s throw at 45° travels about 247 m. That's roughly six times farther than on Earth, because range is inversely proportional to g.

Frequently asked questions

Is 45 degrees always the best angle?

Only when you launch and land at the same height with no air resistance. Launching from above the landing point, as a shot-putter does, makes the best angle lower than 45°. Air resistance also pulls it lower, because high, looping paths spend longest in the air and lose the most speed to drag. Lift from spin, as in golf, changes it again.

What angle gives the maximum height?

Straight up: 90°. Then all of the launch speed goes into climbing, and the maximum height is v² ÷ 2g. A 20 m/s throw on Earth would rise about 20.4 m. The catch is that it lands right back where it started, so the range is zero. Height and range pull in opposite directions, which is why 45° is a compromise.

Does mass affect how far a projectile goes?

Without air resistance, no. Gravity gives every object the same acceleration, 9.81 m/s² on Earth, so a heavy ball and a light one launched at the same speed and angle follow exactly the same path. With air, mass matters: a heavy, compact object is barely slowed by drag, while a light, large one like a beach ball falls far short.

Try it yourself

Open the Projectile Motion simulator and try these:

• Set the speed to 20 m/s and fire at 30°, then at 60°. Compare the two Range readings.
• Fire at 44°, 45° and 46°. Can you beat 45° by even a centimetre?
• Raise the launch height, then hunt for the new best angle. You'll find it below 45°.
• Switch on air resistance, choose the beach ball and watch its path collapse on the way down.
• Swap Earth for the Moon and see how far the same shot goes.

The simulator writes the equations out live as you change things, so you can check every number in the table above. If you'd like to know where the value of g comes from, measure it yourself with a simple pendulum. Projectiles are also a good place to practise Newton's laws: once the ball leaves your hand, gravity is the only force on it (ignoring air), which is exactly why its sideways speed never changes.