PhysicsCalcs
Motion & Kinematics

Projectile Motion Calculator

Calculate range, maximum height, time of flight, launch speed, launch angle, or elevated launch trajectory. Step-by-step solutions and trajectory diagram.

Interactive calculator

Projectile Motion Calculator

Calculate range, max height, time of flight, launch speed, launch angle, or elevated launch trajectory for projectile motion.

Try an example

Launch speed

Angle above horizontal

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Choose: full trajectory, range, height, time, speed, angle, or elevated launch.
  • Enter speed, angle, and gravity (default: Earth).
  • Click Calculate for complete trajectory analysis.

Table of Contents

Key Takeaways

  • Projectile motion combines constant horizontal velocity with vertical free fall.
  • Range R = v₀² sin(2θ) / g is maximised at θ = 45°.
  • Max height H = v₀² sin²θ / (2g).
  • Time of flight T = 2v₀ sinθ / g.
  • Two launch angles (θ and 90° − θ) give the same range.
  • Air resistance significantly affects real projectiles at high speeds.

What Is Projectile Motion?

Projectile motion describes the path of an object launched into the air and subject only to gravity. The motion has two independent components: constant horizontal velocity and uniformly accelerated vertical motion (free fall). The result is a parabolic trajectory.

Projectile Motion Formulas

R=v02sin⁡(2θ)/gR = v_0^2 \sin(2\theta) / g
H=v02sin⁡2θ/2gH = v_0^2 \sin^2\theta / 2g
T=2v0sin⁡θ/gT = 2v_0 \sin\theta / g

Where v₀ is launch speed, θ is launch angle, and g is gravity. These assume flat ground and no air resistance.

Trajectory Diagram

Range (R)Hθv₀

A projectile launched at angle θ follows a parabolic path. Maximum height occurs at the midpoint; range is the horizontal distance to landing.

How to Use the Calculator

  1. Choose the calculation mode for your known values.
  2. Enter speed, angle, distance, and/or launch height.
  3. Click Calculate.
  4. Review range, height, time, and velocity results.

Example Calculations

20 m/s at 45°

R = 400 × 1 / 9.81 = 40.8 m, H = 10.2 m, T = 2.88 s

30 m/s at 30°

R = 900 × 0.866 / 9.81 = 79.5 m

Cliff: 15 m/s, 40°, h = 20 m

Range ≈ 37 m (longer than flat ground)

Common Mistakes

  • Using the wrong angle (from vertical instead of horizontal).
  • Forgetting air resistance for high-speed or long-range problems.
  • Assuming the trajectory is symmetric when launched from a height.
  • Using degrees in trigonometric functions set to radians.
  • Thinking range only depends on speed (angle matters).

Accuracy and Limitations

These formulas assume no air resistance, flat terrain, constant gravity, and no wind. Real projectiles (balls, bullets, rockets) are significantly affected by drag, spin (Magnus effect), altitude, and atmospheric conditions. Use for education and estimation.

FAQ

What is projectile motion?›

Projectile motion is the curved path of an object launched into the air, subject only to gravity. It’s a combination of constant horizontal velocity and uniformly accelerated vertical motion.

Why is 45° the optimal launch angle?›

At 45°, sin(2θ) = sin(90°) = 1, maximising the range formula R = v₀²sin(2θ)/g. This only holds on flat ground in vacuum.

Do two angles really give the same range?›

Yes. For any angle θ < 45°, the complementary angle (90° − θ) gives the same range. The low angle gives a faster, flatter path; the high angle gives a slower, higher arc.

How does air resistance affect projectiles?›

Air drag reduces range (often significantly), lowers the optimal angle below 45°, makes the trajectory asymmetric, and reduces maximum height. Real-world calculations require numerical methods.

What about launching from a height?›

Use the Elevated Launch mode. The projectile has more time in the air, so range and impact speed increase compared to ground-level launch.

Does mass affect projectile range?›

In a vacuum, no. In air, heavier (denser) objects are less affected by drag and travel further.

How do I find the trajectory equation?›

y = x tanθ − gx² / (2v₀²cos²θ). This is a parabola opening downward.

Sources

  1. OpenStax University Physics – Projectile Motion
  2. HyperPhysics – Projectile Motion
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

Learn more about Manish