Projectile Motion Explained — Trajectories, Range and Height

What counts as a projectile

A projectile is any object that is launched and then left to move under gravity alone — a thrown ball after it leaves the hand, a cannonball after it leaves the barrel, water from a fountain. The key idea is that the launch force is over; from that moment, the only force acting is weight pulling straight down.

Air resistance is ignored in the standard treatment, which is a good approximation for heavy, slow objects over short flights and a poor one for feathers, shuttlecocks or high-speed artillery.

Horizontal and vertical motion are independent

This is the single most important idea in the topic. Gravity acts only vertically, so the horizontal component of velocity never changes (no horizontal force), while the vertical component changes at $g = 9.81\ \mathrm{m/s^2}$ downward — exactly like a ball dropped from rest.

Famously, a bullet fired horizontally and a bullet dropped from the same height hit the ground at the same time. The horizontal motion adds no help or hindrance to the fall; the two motions simply coexist.

Split the launch velocity $v_0$ at angle $\theta$ above the horizontal using trigonometry:

$$v_{0x} = v_0 \cos\theta,\qquad v_{0y} = v_0 \sin\theta$$

Time of flight and maximum height

The vertical motion is just constant-acceleration motion with $a = -g$. Taking upward as positive and launching from the ground, the vertical velocity is $v_y = v_0\sin\theta - gt$.

The projectile reaches its peak when $v_y = 0$, giving a time to the top of $t_{up} = v_0\sin\theta / g$. On flat ground the flight is symmetric, so the total flight time is twice that:

$$T = \frac{2v_0\sin\theta}{g}$$

The maximum height comes from $v_y^2 = v_{0y}^2 - 2gH$ with $v_y = 0$ at the top:

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

The range formula

Range is the horizontal distance traveled. Since horizontal velocity is constant, range is simply horizontal speed times flight time:

$$R = v_{0x}T = v_0\cos\theta \cdot \frac{2v_0\sin\theta}{g} = \frac{v_0^2\sin 2\theta}{g}$$

where the identity $2\sin\theta\cos\theta = \sin 2\theta$ was used. This formula assumes launch and landing at the same height and no air resistance.

Because $\sin 2\theta$ peaks at $1$ when $2\theta = 90^\circ$, maximum range occurs at a launch angle of $45^\circ$. Angles symmetric about $45^\circ$ — like $30^\circ$ and $60^\circ$ — give identical ranges.

Worked example: the full trajectory

A soccer ball is kicked at $v_0 = 20\ \mathrm{m/s}$ at $30^\circ$ above the horizontal. Find the range, maximum height and flight time.

First split the velocity: $v_{0x} = 20\cos 30^\circ = 17.32\ \mathrm{m/s}$, $v_{0y} = 20\sin 30^\circ = 10\ \mathrm{m/s}$.

Flight time: $T = 2(10)/9.81 = 2.04\ \mathrm{s}$.

Maximum height: $H = 10^2/(2 \times 9.81) = 5.10\ \mathrm{m}$.

Range: $R = v_{0x}T = 17.32 \times 2.04 = 35.3\ \mathrm{m}$ — matching the direct formula $400\sin 60^\circ/9.81 = 35.3\ \mathrm{m}$.

Launching from a height

The neat formulas above assume landing at launch height. When a projectile is launched from a cliff or a raised platform, the flight lasts longer because the object has extra distance to fall — so the range is larger than $v_0^2\sin 2\theta/g$ predicts.

The fix is to solve the vertical motion separately: with launch height $h_0$, the landing time solves $h_0 + v_{0y}t - \tfrac{1}{2}gt^2 = 0$ (take the positive root), then multiply by the constant horizontal speed $v_{0x}$. There is no single compact range formula for this case, which is why textbooks emphasize the flat-ground version.

Where the ideal model breaks down

Air drag reshapes real trajectories: it shortens range, lowers the peak, and makes the descent steeper than the ascent. For a golf ball, drag and spin (the Magnus effect) matter so much that the vacuum optimum of $45^\circ$ drops to roughly $30^\circ$.

At very high speeds or long ranges, two more corrections appear: the curvature of the Earth changes what 'flat ground' means, and the Coriolis effect deflects long-range artillery. For classroom-scale problems, though, the no-drag parabola is accurate and is the model examiners expect.

Variables and units

Common mistakes

Real-world applications

• Firefighting monitors are aimed using projectile equations to place water streams on upper floors.

• Sports science uses launch-angle analysis to optimize javelin, shot put and long-jump technique.

• Irrigation engineers space sprinkler heads using the range of water jets at given pressures.

• Video-game physics engines integrate the same equations to simulate thrown objects and ballistics.