Trajectory Calculator
Trace a projectile through the air with real drag, see where it lands, and read its position at regular time steps.
☄️ What is a Trajectory Calculator?
A trajectory calculator traces the path of a projectile after it is launched and tells you where it goes, how high it climbs, how long it stays in the air and how fast it is moving when it lands. In the simplest textbook case only gravity acts, and the path is a parabola with tidy closed-form answers. This calculator keeps that case but also lets you switch on air drag, which is what really shapes the flight of a thrown ball, a kicked football, a golf shot or an artillery shell.
Trajectories matter in sport, engineering and teaching. Coaches compare launch angles for a baseball throw or a javelin, engineers estimate where debris, water jets or sprayed material will land, game developers need believable ballistic arcs, and students use trajectories to learn how forces change motion. A launch height matters too: a ball released from a cliff or a rooftop stays in the air longer and travels farther than one launched from the ground at the same speed and angle, and a downward throw from a height can be modelled with a negative launch angle.
With drag the equations have no simple closed form, so the calculator integrates the motion numerically in small time steps with a fourth order Runge-Kutta method and finds the landing point by interpolation. You can pick linear drag, where the force is proportional to speed, or quadratic drag, where the force is half the air density times the drag coefficient times the area times the speed squared. Quadratic drag is the realistic choice for fast objects in air. Both models assume still air, constant density, no spin and flat ground at zero height.
It complements the projectile motion calculator, which covers the ideal vacuum case with textbook formulas. Here you also get a table of position, speed and direction at regular time intervals, plus a chart that overlays the real path on the dashed no-drag parabola, so the effect of air resistance on range, peak height and landing angle is easy to see.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Ideal Throw from the Ground
20 m/s at 45° from ground level, no drag
Example 2 — Launch from a 50 m Cliff
15 m/s at 30° from a height of 50 m, no drag
Example 3 — Baseball Throw with Quadratic Drag
40 m/s at 35° from 1.5 m, mass 0.145 kg, Cd = 0.35, A = 0.0043 m², air 1.225 kg/m³
Example 4 — Linear Drag Teaching Model
20 m/s at 45° from the ground, mass 0.5 kg, linear drag b = 0.1 kg/s
Example 5 — Soccer Ball Kick
25 m/s at 30° from the ground, mass 0.43 kg, Cd = 0.25, A = 0.038 m²
Example 6 — Downward Throw from a Building
10 m/s at −20° (below horizontal) from a height of 30 m, no drag
❓ Frequently Asked Questions
🔗 Related Calculators
How do you calculate the trajectory of a projectile with air resistance?
Integrate the equations of motion numerically: acceleration is gravity plus a drag term opposing the velocity. This calculator uses fourth order Runge-Kutta steps of at most 2 ms and finds the landing point by interpolation, which has no closed form for quadratic drag.
What is the difference between linear and quadratic drag?
Linear drag force is b times speed and applies to slow motion in viscous fluids. Quadratic drag is 0.5 rho Cd A v squared and applies to most balls, bullets and shells in air at high Reynolds number. This page offers both.
How does launch height change a trajectory?
A higher release point adds flight time and range. From 50 m with 15 m/s at 30 degrees and no drag, the flight lasts 4.05 s and covers about 52.59 m, compared with 1.53 s and 19.87 m from the ground.
What angle gives the longest range with drag?
Less than 45 degrees. Drag removes more energy on a long, high path, so the best angle for a drag-affected projectile is typically between 30 and 40 degrees, depending on speed and the drag factor.
Why is the descent steeper than the climb with drag?
Drag slows the horizontal motion during the whole flight while gravity keeps adding downward speed. The path becomes asymmetric and the landing angle is steeper than the launch angle, which the table shows in the direction column.
What drag coefficient should I use for a ball?
A smooth sphere is often modelled with Cd near 0.47, a baseball near 0.3 to 0.4 and a soccer ball near 0.2 to 0.25 at typical speeds. The reference area is the cross-section pi r squared. Treat these as rough values.
How accurate is the numerical integration?
Fourth order Runge-Kutta with a step of at most 2 ms is accurate to better than 0.01 percent for the cases on this page. The no-drag answer matches the analytic formulas to every displayed digit.
What does the position table show?
At each interval it lists time, horizontal distance, height, speed and direction of motion in degrees above the horizontal. Negative directions mean the projectile is descending. The last row is the exact landing point.
Can this model a throw downward from a building?
Yes. Enter a negative angle and a launch height above zero. A 10 m/s throw at -20 degrees from 30 m with no drag lands after 2.15 s and about 20.2 m away.
How is this different from the projectile motion calculator?
The projectile motion calculator gives the ideal vacuum answers only. This trajectory calculator adds optional drag by numerical integration, a table of positions at intervals and a chart that overlays the drag path on the ideal parabola.
Does the calculator include wind, spin or changing air density?
No. It assumes still air of constant density, no spin or lift force and flat ground at height zero. Those effects, such as the Magnus force on a spinning ball, need a more detailed model.