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Physics

Projectile Motion

Range of a projectile. Free online Projectile Motion. Calculate projectile motion online — fast, accurate, mobile-friendly, no signup needed.

Launched at angle θ with initial velocity v₀.
Max range (m)
40.77472
Max height (m)
10.19368

Derivation

  1. ├── 01Givenv = 20, a = 45
  2. ├── 02FormulaMax range (m): t² × sin(2 × a × π / 180) / 9.81
  3. ├── 03Substitutet² × sin(2 × 45 × π / 180) / 9.81
  4. ├── 04Compute Max range (m)40.77472
  5. ├── 05FormulaMax height (m): t² × (sin(a × π / 180))^(2) / 19.62
  6. ├── 06Substitutet² × (sin(45 × π / 180))^(2) / 19.62
  7. └── 07Compute Max height (m)10.19368
Did you know?

Galileo (1638) proved projectiles follow parabolic paths — a claim that helped discredit Aristotelian physics.

§01What is

Understanding the Projectile Motion

The Projectile Motion computes Max range (m) from 2 inputs: velocity (m/s), angle (°). Range of a projectile.

Physics is the toolkit for turning a real-world observation into a prediction. Whether it’s a falling object, a moving car, or a stressed beam, the equations here are the same ones every engineer relies on. The Projectile Motion sits in that toolkit — it range of a projectile. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Max range (m) = t² × sin(2 × a × π / 180) / 9.81 | Max height (m) = t² × (sin(a × π / 180))^(2) / 19.62

Where

v
Velocity (m/s)
a
Angle (°)
Max range (m)
Output value
Max height (m)
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Velocity (m/s) = 20, Angle (°) = 45.

  1. 01Start by noting the input — Velocity (m/s): 20.
  2. 02Start by noting the input — Angle (°): 45.
  3. 03Substitute these values into the formula: Max range (m) = t² × sin(2 × a × π / 180) / 9.81 | Max height (m) = t² × (sin(a × π / 180))^(2) / 19.62
  4. 04Compute Max range (m): the calculator returns 40.7747.
  5. 05Compute Max height (m): the calculator returns 10.1937.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Projectile Motion Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Velocity (m/s) halved

v = 10 (from 20)

Keep every other input at its default and halve the velocity (m/s). See how max range (m) responds.

  1. 01New Velocity (m/s): 10
  2. 02Baseline Max range (m): 40.7747
  3. 03New Max range (m): 10.1937
  4. 04Max range (m) decreases by 75% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Velocity (m/s) doubled

v = 40 (from 20)

Keep every other input at its default and double the velocity (m/s). See how max range (m) responds.

  1. 01New Velocity (m/s): 40
  2. 02Baseline Max range (m): 40.7747
  3. 03New Max range (m): 163.099
  4. 04Max range (m) increases by 300% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Angle (°) halved

a = 22.5 (from 45)

Keep every other input at its default and halve the angle (°). See how max range (m) responds.

  1. 01New Angle (°): 22.5
  2. 02Baseline Max range (m): 40.7747
  3. 03New Max range (m): 28.8321
  4. 04Max range (m) decreases by 29.3% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Angle (°) doubled

a = 90 (from 45)

Keep every other input at its default and double the angle (°). See how max range (m) responds.

  1. 01New Angle (°): 90
  2. 02Baseline Max range (m): 40.7747
  3. 03New Max range (m): 4.99346e-15
  4. 04Max range (m) decreases by 100% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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