Projectile Motion Calculator

Projectile Motion Calculator finds the range, flight time, maximum height and impact speed for any launch speed, angle and height. It also finds the angle or speed to hit a target.

What the Projectile Motion Calculator Solves

Projectile motion is the curved path of an object that is launched and then moves under gravity alone. The Projectile Motion Calculator handles three kinds of problem, chosen under Solve. The first takes a launch speed, an angle and a launch height and returns the whole flight.

The other two work backward from a target. Angle to Hit a Target finds the launch angle for a known speed. Speed to Hit a Target finds the launch speed for a known angle. Both take the distance to the target and its height.

The default launch is 25 m/s at 40° from 1 m above the ground. It lands 63.93 m away after 3.34 s, having peaked at 14.17 m. It hits the ground at 25.39 m/s, at 41.04° below level. A check time of 1.2 s shows the object 13.22 m high and 22.98 m out at that moment.

Projectile Motion Formulas

Every projectile problem starts by splitting the launch speed into two parts. The horizontal part never changes. The vertical part is slowed and then reversed by gravity. OpenStax University Physics builds the whole topic on treating these two motions separately.

$$v_x = v_0 \cos\theta$$

$$v_y = v_0 \sin\theta$$

The time of flight comes from the vertical motion. With a launch height h₀ above the landing ground, it is the positive root of the height equation. The Projectile Motion Calculator uses this general form, so a raised launch needs no special case.

$$t = \frac{v_y + \sqrt{v_y^2 + 2 g h_0}}{g}$$

The range is the horizontal speed times that time, since nothing slows the sideways motion. The maximum height adds the climb to the launch height, and the climb depends only on the vertical part of the speed.

$$R = v_x t$$

$$H = h_0 + \frac{v_y^2}{2 g}$$

From level ground these reduce to the short forms seen in most textbooks. They hold only when the launch and landing heights are equal, so they give the wrong range for a throw released above the ground.

$$R = \frac{v_0^2 \sin 2\theta}{g}$$

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

For anyone who wants a projectile motion calculator with steps, the step-by-step panel writes out each of these lines with the entered numbers. For the default launch it shows v_x = 19.15 m/s and v_y = 16.07 m/s, then the 3.34 s flight time, the 63.93 m range and the 14.17 m peak.

Range and Flight Time by Launch Angle

The table holds the launch speed at 25 m/s from ground level and changes the angle. Range is greatest at 45° and equal for pairs of angles that add up to 90°. Flight time and height keep rising as the launch gets steeper.

The 45° rule only holds from ground level. A raised launch gives the object extra time to fall, so a slightly flatter angle carries farther. The fourth card of the Projectile Motion Calculator reports the best angle for the launch height entered.

Angle or Speed to Hit a Target

When the speed is fixed, there are usually two launch angles that reach the same point. One is a flat arc and the other is a steep one. With x as the distance to the target and y as its height above the launcher, both come from one formula.

$$\tan\theta = \frac{v^2 \pm \sqrt{v^4 – g (g x^2 + 2 y v^2)}}{g x}$$

This is the setting robotics teams use to aim a shooter, which is why people search for an FRC projectile motion calculator. Take a launcher 0.5 m off the floor, a target 2.5 m high and 5 m away, and an exit speed of 10 m/s. The flat arc needs 38.78° and arrives in 0.64 s, moving almost level. The steep arc needs 73.02°, takes 1.71 s and drops in at 68°.

In target problems the third card of the Projectile Motion Calculator gives the minimum speed, which is 8.51 m/s for that target. Below it no angle works, because the term under the square root turns negative. The trajectory graph draws both arcs so the choice between a fast flat shot and a slow falling one is easy to see.

When the angle is fixed, the required speed follows from the same geometry. The angle must point above the straight line to the target, or no speed will get the object there.

$$v = \sqrt{\frac{g x^2}{2 \cos^2\theta \, (x \tan\theta – y)}}$$

The OpenStax chapter has a golfer who wants the ball to land 90 m away, once at 30° and once at 70°. Speed to Hit a Target gives 31.92 m/s for the low shot and 37.06 m/s for the high one, with peaks of 12.99 m and 61.82 m. The textbook’s point is the same, that the 70° shot needs the greater launch speed.

Projectile Motion with Air Resistance

The textbook formulas ignore the air. Under Air Resistance, the Projectile Motion Calculator can include drag on a ball. It uses the drag force from OpenStax, which grows with the square of the speed, and a drag coefficient of 0.45 for a sphere.

$$F_D = \frac{1}{2} \rho C A v^2$$

With drag there is no neat formula for the range, so the flight is stepped through in small slices of time. The table runs five balls through the Projectile Motion Calculator at 25 m/s and 40° from 1 m in sea-level air. Without air, every one of them would land at 63.93 m.

What matters is mass compared with frontal area. The heavy shot barely notices the air, while the light table tennis ball loses four fifths of its range. Every row uses the smooth-sphere coefficient, so real balls with seams, fuzz or dimples will differ somewhat.

Drag also changes the shape of the path. The baseball peaks at 12.08 m, down from 14.17 m, and comes down at 49.18°, steeper than the 40° it left at.

Thinner air helps a little. At 2,000 m, where the density is 1.007 kg/m³, the same throw reaches 49.24 m. A straight drop with drag is covered by the Free Fall with Air Resistance Calculator.

Horizontal and Vertical Launches

A horizontal projectile is the 0° case. It needs a launch height, since it starts falling at once. At 25 m/s from 10 m up, the object lands 35.7 m out after 1.43 s and hits at 28.66 m/s. The Horizontal Projectile Motion Calculator works this case backward from a range or a flight time as well.

A vertical projectile is the 90° case. The same 25 m/s from 1 m rises to 32.87 m, spends 5.14 s in the air and lands where it started. Peak-only problems, including ones that start from a range or a target height, are quicker in the Maximum Height Calculator.

Gravity in the Projectile Motion Calculator can be switched to other worlds. The default launch that travels 63.93 m on Earth goes 167.54 m on Mars and 385.88 m on the Moon, but only 27.79 m on Jupiter. A launch from a ramp onto a landing at a different height is the job of the Car Jump Distance Calculator.

Graphing the Path in Desmos

The path of a projectile without air is a parabola. Removing time from the two motions gives height y as a function of distance x, which can be typed straight into Desmos or any graphing tool.

$$y = h_0 + x \tan\theta – \frac{g x^2}{2 v_0^2 \cos^2\theta}$$

For the default launch this becomes y = 1 + 0.8391x − 0.01337x². The curve crosses y = 0 at x = 63.93, which is the range. The trajectory table in the tool lists matching points, such as 13.72 m high at 25.57 m out.