Projectile Motion Explained: Range, Max Height and Why 45 Degrees Wins

Throw a ball, fire a cannon, kick a football, or aim a fountain nozzle, and the path traced through the air is the same shape every time: a parabola. The whole subject of projectile motion comes down to one quiet trick that makes the math tractable. Instead of chasing the curved path directly, you split the motion into two pieces — one horizontal, one vertical — solve each on its own, and stitch the answers back together at the end.

This article walks through how that split works, how to get range, maximum height and flight time out of a launch speed and angle, and why 45 degrees is the famous magic number for distance on flat ground. Along the way you can check every claim yourself in the projectile motion calculator, which solves the full set of equations and draws the trajectory to scale.

The one idea that makes it work: independent components

A projectile leaves your hand at speed v₀ pointed at angle θ above the horizontal. The first move is always the same: break that velocity into a horizontal part and a vertical part.

• Horizontal velocity: vₓ = v₀ · cos θ
• Vertical velocity: v_y = v₀ · sin θ

Here is the part that surprises most students the first time they see it: the horizontal and vertical motions are completely independent of each other. Gravity pulls straight down, so it only ever touches the vertical motion. The horizontal motion has no force acting on it (we are ignoring air for now), so vₓ never changes from launch to landing. The projectile drifts sideways at a constant rate while it accelerates downward — and neither half cares what the other is doing.

That is why a bullet fired horizontally and a bullet dropped from the same height hit the ground at the same moment. They share the same vertical story (start with zero vertical speed, fall under gravity), so their flight times match exactly, even though one travels half a kilometer sideways and the other lands at your feet.

If splitting a vector into components feels rusty, a quick pass through a trigonometry calculator makes the cos θ and sin θ steps concrete before you plug them into the kinematics.

From components to flight time and max height

Once you have vₓ and v_y, the rest falls out of two standard kinematics relations.

Flight time comes from the vertical motion alone. For a launch and landing at the same height, the projectile goes up, stops, and comes back down symmetrically, so:

t = 2 · v_y / g = 2 · v₀ · sin θ / g

where g is gravitational acceleration (9.81 m/s² on Earth).

Maximum height is reached when the vertical velocity hits zero — the top of the arc:

h_max = v_y² / (2g) = (v₀ · sin θ)² / (2g)

Horizontal range is just constant horizontal speed multiplied by the time in the air:

R = vₓ · t = (v₀ · cos θ) · (2 · v₀ · sin θ / g)

Using the identity 2 · sin θ · cos θ = sin(2θ), that whole expression collapses into the cleanest formula in the subject:

R = v₀² · sin(2θ) / g

Notice that range scales with the square of the launch speed but only linearly through the sine term for angle. Double the speed and you quadruple the distance; the angle just steers how that distance gets spent.

A worked example: 20 m/s at 45 degrees

Let me run a concrete shot end to end, because the formulas only click once you watch numbers move through them. Launch a projectile at v₀ = 20 m/s and θ = 45° from flat ground, g = 9.81 m/s².

First, the components:

• vₓ = 20 · cos 45° = 20 · 0.7071 ≈ 14.14 m/s
• v_y = 20 · sin 45° = 20 · 0.7071 ≈ 14.14 m/s

At exactly 45 degrees the two components are equal — the throw splits its speed evenly between going forward and going up.

Flight time:

t = 2 · 14.14 / 9.81 ≈ 2.88 s

Maximum height:

h_max = 14.14² / (2 · 9.81) ≈ 200 / 19.62 ≈ 10.2 m

Range, the headline number:

R = 20² · sin(90°) / 9.81 = 400 · 1 / 9.81 ≈ 40.8 m

So a 20 m/s launch at 45 degrees travels about 40.8 meters, peaks a little over 10 meters up, and stays airborne for roughly 2.9 seconds. Drop those exact inputs into the projectile motion calculator and you will see the same figures, plus a parabola marking the apex at 10.2 m and the landing point at 40.8 m.

Why 45 degrees gives the maximum range

Look again at R = v₀² · sin(2θ) / g. The launch speed and gravity are fixed for a given throw, so the only thing you can tune is the angle, and it enters purely through sin(2θ). The sine function maxes out at 1, and it reaches that peak when its argument is 90 degrees. That means:

2θ = 90°   →   θ = 45°

At 45 degrees, sin(2θ) = sin(90°) = 1, and the range hits its largest possible value of v₀²/g. Tilt away from 45 in either direction and sin(2θ) drops below 1, shrinking the range.

This also explains a neat symmetry: 30 degrees and 60 degrees give exactly the same range, because sin(60°) and sin(120°) are equal. Any pair of angles that adds up to 90 degrees lands the same distance away — one as a low, fast, flat shot, the other as a high, slow lob. They just take very different amounts of time and reach very different heights getting there.

The intuition behind 45 being the sweet spot: range is the product of two competing factors. A flatter angle gives more horizontal speed but less hang time; a steeper angle gives more hang time but less horizontal speed. Forty-five degrees is the exact balance point where the gain in one stops outweighing the loss in the other.

When 45 degrees stops being the answer

Every result above assumed the projectile lands at the same height it left from. The moment you launch from a cliff, a raised platform, or a thrower's shoulder, that assumption breaks — and so does the 45-degree rule.

Launch from a height and gravity gets extra time to carry the projectile forward before it finally hits the lower ground. A flatter, faster-horizontal shot now exploits that bonus time better than a steep one, so the optimal angle drops below 45 degrees. The exact optimum is:

θ = arctan( v₀ / √(v₀² + 2gh) )

For v₀ = 20 m/s off a 10 m ledge on Earth, that works out to about 38 degrees, not 45. Air resistance pushes the real-world optimum even lower, which is part of why long jumpers, shot putters, and discus throwers all release well under 45 degrees in practice.

The vacuum model here is the correct textbook baseline — accurate to within a few percent for dense, slow objects over short ranges, and the answer your physics homework expects. Treat it as an upper bound on real distance, then shade your expectations down for anything light or fast.

If you want to dig deeper into the geometry of those right-triangle velocity components, the trigonometry calculator pairs naturally with the projectile tool: one shows you the angles and sides, the other shows you where the shot lands.

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Made by Toolora · Updated 2026-06-13