Projectile Motion on an Incline

Last updated: 2026-04-19

Projectile motion on an incline

Projectile motion on an incline is the trajectory of a body launched from a point on a sloped surface, where the launch point and the landing point lie on the same inclined plane rather than on a common horizontal level. The standard projectile problem assumes the projectile returns to the height it started from; introducing a tilted landing surface changes the flight time, the range, and the launch angle that maximizes range.

This calculator solves projectile motion onto an inclined plane that passes through the launch point. A positive incline angle means the landing surface rises away from the launcher (uphill landing); a negative incline angle means it falls away (downhill landing).

How it works

Coordinates are placed at the launch point with $x$ horizontal and $y$ vertical. With launch speed $v_0$ and launch angle $\theta$ measured from the horizontal, the path is the standard parabola:

$x(t) = v_0 \cos\theta \cdot t \qquad y(t) = v_0 \sin\theta \cdot t - \frac{1}{2} g t^2$

The incline is a straight line of slope $\tan\alpha$ through the origin:

$y\_{\text{incline}}(x) = x \tan\alpha$

The projectile lands where the trajectory meets the incline. Setting $y(t) = x(t)\tan\alpha$ and solving for $t$ gives the flight time:

$t_f = \frac{2 v_0 \sin(\theta - \alpha)}{g \cos\alpha}$

The range measured along the incline follows from substituting $t_f$ back into the path:

$R\_{\text{incline}} = \frac{2 v_0^2 \cos\theta \sin(\theta - \alpha)}{g \cos^2\alpha}$

Optimum launch angle

The launch angle that maximizes the range along the incline depends on the slope:

$\theta\_{\text{opt}} = 45° + \frac{\alpha}{2}$

For flat ground ($\alpha = 0$) this reduces to the familiar 45°. The half-angle term bisects the slope direction and the vertical, so a steeper uphill incline raises the optimum angle and a downhill incline lowers it. The table below lists the optimum angle across a range of slopes:

Worked example

Consider a projectile launched at $v_0 = 20$ m/s and $\theta = 55°$ up a slope of $\alpha = 20°$, with $g = 9.81\ \text{m/s}^2$. The launch angle equals the optimum $45° + \alpha/2 = 55°$, so this case sits at maximum range along the slope. The flight time is

$t_f = \frac{2 (20) \sin(55° - 20°)}{9.81 \cos 20°} = \frac{40 \sin 35°}{9.81 \cos 20°} \approx 2.49\ \text{s}$

and the range measured along the incline is

$R\_{\text{incline}} = \frac{2 (20)^2 \cos 55° \sin 35°}{9.81 \cos^2 20°} \approx 30.4\ \text{m}$

The horizontal projection of that landing point is $R = R\_{\text{incline}} \cos\alpha \approx 28.6$ m.

Uphill, downhill, and the angle constraint

The same formulas cover both directions through the sign of $\alpha$. An uphill slope of 20° gives an optimum launch angle of 55°, while a downhill slope of −20° gives 35°. A downhill incline also lengthens the flight time relative to flat ground at the same launch angle, because the projectile falls past the launch height before landing.

For uphill shots the launch angle must exceed the incline angle. When $\theta \leq \alpha$ the projectile is moving parallel to or into the slope at launch, $\sin(\theta - \alpha)$ is zero or negative, and the model returns a degenerate (zero or negative) range.

Applications and limits

The inclined-plane case is the general form of which flat-ground projectile motion is a special instance, which makes it the standard model for several physical situations:

• Ski jumping. The landing knoll on world-class hills slopes roughly 30–37° downhill. With a downhill landing the range-optimizing take-off angle is well below 45°, consistent with the near-flat trajectories modern technique produces.
• Mountain artillery. Firing tables for indirect fire across sloped terrain encoded slope corrections: a weapon firing uphill at a target needs a higher launch angle than the same weapon on level ground.
• Throws over sloped ground. A throw down a hillside has a longer flight time and a flatter optimum angle than the same throw on level ground.

The model has the following limits:

• No air resistance. The vacuum model overstates range. Drag depends on speed, projectile geometry, and air density.
• The incline passes through the launch point. If the slope begins at a different elevation, treat the launch point as lying on the slope or use a separate model.
• First impact only. The range along the incline is the distance to the first landing point. Bouncing, rolling, sliding, and shattering after impact are not modelled.