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Projectile Motion Calculator

Physics

Active gravity:

9.81 m/s²

About This Tool

🏹 Projectile Motion Calculator – Trajectory, Range & Height

The Projectile Motion Calculator is a free online physics tool that instantly computes the complete trajectory of any projectile launched at an angle. Enter the initial velocity, launch angle, optional starting height, and gravitational acceleration to obtain time of flight, maximum height, horizontal range, velocity components, impact speed, and a fully interactive parabolic trajectory graph.

Whether you are a student solving kinematics problems, an educator demonstrating classical mechanics, or just curious about how a ball travels through the air, this free projectile motion simulator gives you instant answers with step-by-step formula breakdowns.

📘 What Is Projectile Motion?

Projectile motion describes the movement of an object launched into the air under the sole influence of gravity (assuming no air resistance). The key insight is that horizontal and vertical motion are independent of each other:

  • Horizontal motion — constant velocity, no acceleration.
  • Vertical motion — uniformly accelerated (or decelerated) by gravity.

The combination of constant horizontal motion and uniformly accelerated vertical motion produces the characteristic parabolic path every projectile follows.

⚙️ How the Projectile Motion Calculator Works

Given an initial speed v, launch angle θ, starting height h₀, and gravitational acceleration g, the calculator applies these classical equations:

Velocity decomposition:

vx  = v · cos(θ)   — horizontal component (constant)
vy₀ = v · sin(θ)   — initial vertical component

Time of flight (solving y = 0 for the general elevated case):

T = (vy₀ + √(vy₀² + 2·g·h₀)) / g

For a ground-level launch (h₀ = 0) this simplifies to T = 2·vy₀ / g.

Maximum height above the ground:

H_max = h₀ + vy₀² / (2·g)

Horizontal range:

R = vx · T

Apex time (time to reach the highest point):

t_apex = vy₀ / g

🧮 Practical Examples

Example 1 — Classic 45° ground launch:
Initial velocity = 20 m/s, angle = 45°, height = 0 m, g = 9.81 m/s²

vx = 20 · cos(45°) ≈ 14.14 m/s
vy₀ = 20 · sin(45°) ≈ 14.14 m/s
T = 2 × 14.14 / 9.81 ≈ 2.88 s
H_max = 14.14² / (2 × 9.81) ≈ 10.19 m
R = 14.14 × 2.88 ≈ 40.73 m

Example 2 — Elevated launch from 10 m:
Initial velocity = 15 m/s, angle = 30°, height = 10 m, g = 9.81 m/s²

vx = 15 · cos(30°) ≈ 12.99 m/s
vy₀ = 15 · sin(30°) = 7.50 m/s
T = (7.50 + √(7.50² + 2×9.81×10)) / 9.81 ≈ 2.29 s
H_max = 10 + 7.50² / (2×9.81) ≈ 12.87 m
R = 12.99 × 2.29 ≈ 29.74 m

Example 3 — Horizontal launch (angle = 0°) from 20 m:
An object fired horizontally at 10 m/s from a 20 m cliff.

vx = 10 m/s, vy₀ = 0
T = √(2 × 20 / 9.81) ≈ 2.02 s
R = 10 × 2.02 ≈ 20.2 m

💡 Tips and Best Practices

  • 45° for maximum range — on a flat surface, launching at 45° always gives the longest range. Complementary angles (e.g., 30° and 60°) produce the same range.
  • Planet presets — switch gravity to Moon (1.62 m/s²) or Mars (3.72 m/s²) to see how the same throw would behave on another world. On the Moon, the same launch reaches roughly 6× the range it would on Earth.
  • Elevated launches — launching from a height always extends the range beyond the flat-ground prediction. Use the initial height field for cliff, ramp, or table-top scenarios.
  • Real-world corrections — this tool uses ideal equations without air resistance. For dense, slow-moving objects the results are a good approximation, but for bullets or golf balls at high speed, drag will noticeably reduce range and height.
  • Export options — use the Copy button to paste results into a report, or download a CSV for spreadsheet analysis or lab write-ups.

🔗 Related Physics Concepts

Projectile motion is closely linked to several other areas of classical mechanics. Kinetic energy and potential energy exchange continuously along the trajectory — at the apex all kinetic energy from the vertical component has converted to gravitational potential energy. The acceleration calculator covers uniform acceleration in one dimension, while the gravitational force calculator helps you determine the exact value of g for non-standard bodies. For objects on slopes, the inclined plane calculator provides complementary launch-angle analysis. Together these tools cover the complete toolkit of Newtonian kinematics and dynamics.

Frequently Asked Questions

Is the Projectile Motion Calculator free?

Yes, Projectile Motion Calculator is totally free :)

Can I use the Projectile Motion Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Projectile Motion Calculator?

Yes, any data related to Projectile Motion Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is projectile motion?

Projectile motion is the curved path an object follows when launched into the air under gravity alone. The horizontal velocity stays constant while gravity continuously accelerates the object downward, producing a parabolic trajectory.

How does the Projectile Motion Calculator work?

Enter the initial velocity, launch angle, optional initial height, and gravity. The calculator decomposes velocity into horizontal (vx = v·cosθ) and vertical (vy = v·sinθ) components, then uses kinematic equations to compute time of flight, maximum height, range, and plots the full trajectory.

What launch angle gives the maximum horizontal range?

For a launch from ground level, 45° yields the maximum range. When launching from an elevated height, the optimal angle drops slightly below 45°. This relationship only holds for ideal motion without air resistance.

How is time of flight calculated for an elevated launch?

For a launch from height h, the flight time is found by solving 0 = h + vy·t − ½g·t², giving T = (vy + √(vy² + 2gh)) / g. For ground launches (h = 0) this simplifies to T = 2·vy / g.

Can I simulate projectile motion on other planets?

Yes. Use the gravity preset to select Moon (1.62 m/s²), Mars (3.72 m/s²), Jupiter (24.79 m/s²), or enter a completely custom value. This lets you compare how the same launch would behave across different celestial bodies.

Does this calculator account for air resistance?

No. The calculator uses ideal projectile motion equations that assume no air resistance. In practice, drag reduces both range and maximum height, especially at high speeds. Results here represent the theoretical maximum under ideal conditions.