Projectile motion calculator

The formula

On level ground (launch height zero):

$$R = \frac{v_0^2 \sin 2\theta}{g}$$ $$H = \frac{(v_0 \sin\theta)^2}{2g}$$ $$T = \frac{2 v_0 \sin\theta}{g}$$

$v_0$ is the launch speed, $\theta$ the angle above horizontal, $g$ the acceleration due to gravity, $R$ the range, $H$ the peak height and $T$ the time of flight. With a launch height $h_0$, the calculator uses the general form derived below.

Where it comes from

The trick is to split the motion into two directions that don’t affect each other. Gravity pulls only downward, so:

• Horizontally, nothing changes the speed: $v_x = v_0\cos\theta$, constant.
• Vertically, the projectile starts at $v_y = v_0\sin\theta$ and slows by $g$ every second, exactly like a ball thrown straight up.

The vertical position after time $t$ follows from $s = ut + \tfrac{1}{2}at^2$ with $a = -g$:

$$y = h_0 + v_0\sin\theta\,t - \tfrac{1}{2}g t^2$$

The projectile lands when $y = 0$. That’s a quadratic in $t$, and its positive root is the flight time:

$$T = \frac{v_0\sin\theta + \sqrt{(v_0\sin\theta)^2 + 2 g h_0}}{g}$$

With $h_0 = 0$ the square root becomes $v_0\sin\theta$ and $T = 2v_0\sin\theta / g$.

The range is horizontal speed times flight time:

$$R = v_0\cos\theta \cdot T$$

On level ground that’s $2v_0^2\sin\theta\cos\theta / g$, and the identity $2\sin\theta\cos\theta = \sin 2\theta$ gives the tidy form above.

At the peak, vertical velocity is zero. From $v^2 = u^2 + 2as$, the height gained is $(v_0\sin\theta)^2 / 2g$, which is added to $h_0$.

The landing speed comes from energy: kinetic energy at launch plus the energy from falling $h_0$, so $v = \sqrt{v_0^2 + 2gh_0}$.

Worked example

A ball kicked at 20 m/s and 45° on flat ground:

$$T = \frac{2 \times 20 \times 0.7071}{9.81} = 2.88\ \text{s}$$ $$H = \frac{(14.14)^2}{2 \times 9.81} = 10.2\ \text{m}$$ $$R = \frac{20^2 \times \sin 90^\circ}{9.81} = 40.8\ \text{m}$$

Watch out for

• 45° is best only on level ground in a vacuum. From a height, a lower angle goes farther. With air drag, the best angle for a real ball is usually 35° to 42°.
• Air resistance matters at speed. A golf ball or a baseball can travel well under half of the vacuum range. Treat these answers as an upper limit for fast, light objects.
• Complementary angles share a range. On level ground, 30° and 60° land in the same spot, because $\sin 60^\circ = \sin 120^\circ$. The steeper shot just takes longer and goes higher.

Sources

• OpenStax, University Physics Volume 1, section 4.3 "Projectile Motion"

Last checked November 11, 2026