Trajectory Calculator plots the flight path of a projectile and writes its trajectory formula, checks the height at any distance against a set target and adds a path with air drag.
Velocity, angle and height set the trajectory. The first panel checks the height at any distance x.
Check Point, Landing and Gravity
Air Resistance
Step-by-Step Working
- Known values in m and m/s: v₀ = 31.9 m/s, α = 30°, h = 0 m, g = 9.80665 m/s²
- Velocity components: vₓ = v₀ × cos(α) = 27.6262 m/s, v_y = v₀ × sin(α) = 15.95 m/s
- Formula terms: tan(α) = 0.57735, g ÷ 2 = 4.90332, (v₀ × cos α)² = 27.6262² = 763.208
- Trajectory formula: y = 0 + 0.5774x − 4.903x² / 763.21
- Divide out the last term: 4.90332 ÷ 763.208 = 0.00642463, so y = 0.5774x − 0.006425x²
- Height at x = 60 m: y = 0 + 0.57735 × 60 − 0.00642463 × 60² = 11.5124 m
- Maximum height (vertex): x = vₓ × v_y ÷ g = 44.9326 m, y = h + v_y² ÷ (2 × g) = 12.9709 m
- Horizontal range: set y = 0 m and take the positive root, x = 89.8652 m
Trajectory Points Table
| Distance (x) | Height (y) | y With Drag | Time (t) |
|---|---|---|---|
| 0 m | 0 m | 0 m | 0 s |
| 8.99 m | 4.67 m | 4.65 m | 0.3253 s |
| 17.97 m | 8.30 m | 8.14 m | 0.6506 s |
| 26.96 m | 10.90 m | 10.33 m | 0.9759 s |
| 35.95 m | 12.45 m | 11.10 m | 1.30 s |
| 44.93 m | 12.97 m | 10.28 m | 1.63 s |
| 53.92 m | 12.45 m | 7.72 m | 1.95 s |
| 60.00 m | 11.51 m | 4.90 m | 2.17 s |
| 62.91 m | 10.90 m | 3.22 m | 2.28 s |
| 71.89 m | 8.30 m | Landed | 2.60 s |
| 80.88 m | 4.67 m | Landed | 2.93 s |
| 89.87 m | 0 m | Landed | 3.25 s |
What the Trajectory Calculator Shows
A trajectory is the path a thrown or launched object follows through the air. The Trajectory Calculator draws that flight path from three values, the initial velocity, the angle of launch and the initial height. It also writes the path as a formula, with your numbers already in it.
Four result cards then answer the usual follow-up questions. They give the height at any distance you choose, the velocity and angles needed to hit a target, the range and maximum height, and the same launch with air drag. The tool opens in a phone browser, so no app download is needed.
Trajectory Formula of a Projectile
Without air resistance, a projectile moves sideways at a steady speed and falls under gravity at the same time. Removing time from those two motions leaves one equation that gives the height y at any horizontal distance x. This is the general equation of trajectory.
$$y = h + x \tan\alpha – \frac{g x^2}{2 v_0^2 \cos^2\alpha}$$
Here h is the initial height, α is the angle of launch, v0 is the initial velocity and g is gravity, 9.80665 m/s² on Earth. The same result appears in OpenStax University Physics for a launch from ground level. The curve is a parabola that opens downward.
The last term is always subtracted, because gravity pulls the path below the straight launch line. Some published versions print a plus sign there, which would send the projectile climbing forever. A launch at 10 m/s and 30° follows y = 0.5774x − 0.06538x², landing 8.83 m away after peaking at 1.27 m.
Worked Example at 31.9 m/s and 30 Degrees
The Trajectory Calculator opens with a launch at 31.9 m/s and 30° from ground level. The velocity splits into 27.63 m/s across and 15.95 m/s up. With tan 30° = 0.5774 and (v0 cos α)² = 763.21, the trajectory formula becomes y = 0.5774x − 4.903x² ÷ 763.21.
Dividing out the last term gives the short form, y = 0.5774x − 0.006425x². Put in x = 60 m and the height is 0.5774 × 60 − 0.006425 × 3,600 = 11.51 m. The projectile is 2.17 seconds into its flight at that point.
The peak is the vertex of the parabola. It sits 44.93 m from the launch point at a maximum height of 12.97 m. The path returns to the ground at 89.87 m, which is the horizontal range, after 3.25 seconds in the air.
The Trajectory Points Table lists twelve points along the path with the time at each one. For the peak, the flight time or the range on their own, see the Maximum Height Calculator, the Time of Flight Calculator and the Projectile Range Calculator.
Height at a Distance and Hitting a Target
The first card turns the tool into a height trajectory calculator. Enter a check distance and a target height in the lower section, such as a wall, a crossbar or a hoop. The card turns green when the path passes above the target and red when it passes below.
In the example the target is 10 m high at 60 m. The path is at 11.51 m there, so it clears the target by 1.51 m. If the projectile lands before the check distance, the card shows how far short it falls.
The second card of the Trajectory Calculator turns the question around and finds the launch that passes exactly through the target point. Keeping the 30° angle, the initial velocity must be 30.91 m/s. Keeping the 31.9 m/s velocity, two angles work, 28.37° and 71.09°.
$$\tan\theta = \frac{v_0^2 \pm \sqrt{v_0^4 – g(g x^2 + 2 y v_0^2)}}{g x}$$
The two signs give a flat path and a steep one that cross at the same point. If the square root is negative, the target is out of reach at that velocity. The card also gives the minimum initial velocity that can reach the point at any angle, 26.35 m/s here.
Trajectory Calculator With Drag
Real projectiles fall short of the ideal parabola. The fourth card and the dashed curve on the plot repeat the launch with a drag force that grows with the square of the velocity. The Trajectory Calculator steps through the flight in small time intervals, because this case has no simple formula.
| Projectile and air | Range with drag | Range lost |
|---|---|---|
| No air, ideal path | 89.87 m | None |
| Baseball at sea level | 67.63 m | 22.24 m (24.7%) |
| Baseball at 3,000 m | 72.02 m | 17.85 m (19.9%) |
| Soccer ball at sea level | 50.79 m | 39.07 m (43.5%) |
All four rows use the same 31.9 m/s launch at 30°. Drag also lowers the path along the way. At 60 m the baseball is 6.61 m below the ideal curve, and its peak drops from 12.97 m to 11.10 m. The soccer ball loses more because it is large for its mass.
Both presets use NASA Glenn figures. The baseball is 5 ounces and 2.9 inches across with a drag coefficient of 0.3. The soccer ball is one pound and 0.75 feet across with a drag coefficient of 0.25. Choose Custom object to enter your own mass, diameter and coefficient.
The drag model has no spin and no lift, and those change real ball flights a great deal. For batted and pitched baseballs with spin, physicist Alan Nathan publishes a spreadsheet at the University of Illinois. The Terminal Velocity Calculator covers the drag force on a falling object.
Arrow and Bullet Trajectories
The Trajectory Calculator gives a first estimate for an arrow from the same parabola. Set the velocity unit to ft/s and the result units to US. An arrow leaving level at 280 ft/s from 5 ft up is at 4.26 ft by 20 yards, a drop of 8.86 inches. The Arrow Speed Calculator supplies the launch speed from a bow’s rating.
The formula for bullet trajectory starts the same way. For a level shot the drop below the bore line is g x² ÷ (2 v0²). At 300 m/s that is 13.6 cm at 50 m. Real bullets slow down quickly from drag, so drop grows faster than this at long range.
Bullet drag is described by a ballistic coefficient, not by one fixed drag coefficient, so the drag card here is not a bullet trajectory calculator. The Ballistic Coefficient Calculator and the MOA Calculator handle that side of shooting.
Range, Angle and Radius of a Trajectory
How do you find the range from the equation of trajectory?
Set y to zero and solve for x. For a ground-level launch the short form is y = bx − ax², so the path returns to the ground at x = b ÷ a. In the example that is 0.5774 ÷ 0.006425 = 89.87 m. Written out, the same step gives the range formula R = v0² sin 2α ÷ g.
What are the three formulas of projectile motion?
They are the time of flight, the maximum height and the range, all for a launch from ground level. Time of flight is T = 2 v0 sin α ÷ g. Maximum height is H = v0² sin² α ÷ (2g). Range is R = v0² sin 2α ÷ g.
For 31.9 m/s at 30° they give 3.25 s, 12.97 m and 89.87 m, the same values the Trajectory Calculator shows. The Projectile Motion Calculator works with all three at once and can solve for the angle or speed.
How do you calculate the flight path angle?
The flight path angle is the direction of the velocity at a point on the path. The sideways velocity stays at v0 cos α, and the vertical velocity has fallen to v0 sin α − g x ÷ (v0 cos α). The angle is the arctangent of vertical over sideways velocity.
The first card reports it as Velocity Angle at x. At 60 m in the example it is −10.96°, so the projectile is already descending. The angle equals the launch angle at the start, zero at the peak, and the negative of the launch angle on landing at launch height.
How do you find the radius of a trajectory?
A parabola has no single radius, but its curvature at the peak is easy to find. There the velocity is purely sideways and gravity acts straight across it, so the radius of curvature is (v0 cos α)² ÷ g. For the example launch that is 763.21 ÷ 9.80665 = 77.83 m.