Free Projectile Range Calculator

About the Projectile Range Calculator

The Projectile Range Calculator is a free online tool for quickly determining the horizontal displacement of an object undergoing projectile motion. As a dedicated physics projectile range tool, it requires only three inputs —initial velocity, launch angle, and launch height— to compute the range of projectile. This makes it an ideal horizontal displacement calculator for students, teachers, and anyone working in physics or ballistics. The calculations assume a vacuum (no air resistance) to focus on ideal projectile motion.

Projectile Range Formulas

The formula for the range of a projectile depends on whether the launch occurs from ground level (zero initial height) or from an elevated position.

Launch from the Ground (Initial Height = 0)

When an object is launched from the ground and lands at the same elevation, the vertical displacement at landing is zero. Starting from the kinematic equation for vertical motion:

$$0 = V_0 \sin(\alpha) \cdot t - \frac{1}{2} g t^{2}$$

Solving for the time of flight $t$ gives:

$$t = \frac{2 V_0 \sin(\alpha)}{g}$$

The horizontal velocity remains constant at $V_0 \cos(\alpha)$, so the horizontal range $d$ is:

$$d = V_0 \cos(\alpha) \cdot t = V_0 \cos(\alpha) \cdot \frac{2 V_0 \sin(\alpha)}{g} = \frac{2 V_0^{2} \sin(\alpha) \cos(\alpha)}{g}$$

Using the identity $\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$, the formula simplifies to:

$$d = \frac{V_0^{2} \sin(2\alpha)}{g}$$

From this expression, the maximum range occurs when $\sin(2\alpha) = 1$, i.e., $2\alpha = 90^\circ$ or $\alpha = 45^\circ$.

Launch from an Elevation (Initial Height > 0)

If the projectile is launched from a height $h$ above the landing point, the vertical motion equation becomes:

$$h + V_0 \sin(\alpha) \cdot t - \frac{1}{2} g t^{2} = 0$$

Solving this quadratic for $t$ yields the time of flight:

$$t = \frac{V_0 \sin(\alpha) + \sqrt{\left(V_0 \sin(\alpha)\right)^{2} + 2 g h}}{g}$$

The horizontal range is then:

$$d = V_0 \cos(\alpha) \cdot t = V_0 \cos(\alpha) \cdot \frac{V_0 \sin(\alpha) + \sqrt{\left(V_0 \sin(\alpha)\right)^{2} + 2 g h}}{g}$$

This is the general formula used by the projectile range calculator when a non‑zero initial height is provided.

Example: Volcanic Ejection

Consider a rock ejected from a volcano with an initial speed of $30 \, \text{m/s}$ at an angle of $25^\circ$. The launch point is 100 m higher than the landing area. Using the elevated launch formula:

$$d = 30 \cos(25^\circ) \cdot \frac{30 \sin(25^\circ) + \sqrt{(30 \sin(25^\circ))^2 + 2 \cdot 9.81 \cdot 100}}{9.81} \approx 162.87 \, \text{m}$$

The calculator instantly provides this result, demonstrating how even a moderate launch angle and height can produce a substantial horizontal reach.

Important Considerations

• Air resistance is neglected in all calculations. In reality, drag reduces the projectile range, and its effect depends on the object’s shape, cross‑section, and surface properties.
• The initial height greatly influences the range; a higher launch point gives a longer flight time and a larger horizontal displacement, assuming the same initial speed and angle.
• The calculator accepts any consistent units (meters, feet, etc.), but the standard unit for range is meters in physics contexts.