Answer first
What this calculator tells you
Calculate the range, maximum height and flight time of a projectile from its launch speed and angle. Predict where a thrown or launched object lands on level ground. Formula: Range = v² × sin 2θ ÷ g; maximum height = v² × sin² θ ÷ (2g); flight time = 2v × sin θ ÷ g, on level ground with no air resistance. At the worked-example inputs, the range (m) is 40.8. Holding every other input steady, moving launch speed (m/s) from 16 to 24 moves the result from 26.1 to 58.7.
Transparent method
The formula
Predict where a thrown or launched object lands on level ground.
Worked example
Example inputs
How to interpret the result
A launched object rises and falls under gravity while it drifts forward at a steady pace. At 20 meters per second and 45 degrees, it flies about 40.8 meters, peaks near 10.2 meters and spends about 2.9 seconds in the air. On level ground with no air resistance, 45 degrees gives the longest range, and angles equally far from it above and below give the same distance.
At the worked-example inputs the range (m) is 40.8. It rises with launch speed (m/s) and falls as gravity (m/s²) and launch angle (degrees) increase.
These are textbook formulas for ideal conditions: no air resistance, no friction, gravity of 9.81 meters per second squared and gases that behave ideally. Real results differ, so treat them as first estimates and use the units the formulas expect (meters, kilograms, seconds).
Before you rely on it
What to check
Remember the model leaves out air resistance and assumes level ground. A real throw lands shorter, and the best angle is a little lower than 45.
The common error
Where people go wrong with projectile motion calculator
Entering the speed in miles per hour or feet per second. The formulas use meters and seconds, and mixed units give a range that is far off.
Sensitivity evidence
How launch speed (m/s) changes the range (m)
Holding every other input at the worked-example value, moving launch speed (m/s) from 16 to 24 moves the range (m) from 26.1 to 58.7: a spread of 32.6, or 80% of the worked-example result.
| Launch speed (m/s) | Range (m) | Maximum height (m) | Flight time (s) |
|---|---|---|---|
| 16 | 26.1 | 6.5 | 2.3 |
| 18 | 33 | 8.3 | 2.6 |
| 20worked example | 40.8 | 10.2 | 2.9 |
| 22 | 49.3 | 12.3 | 3.2 |
| 24 | 58.7 | 14.7 | 3.5 |
Every input, tested
Which input moves the range (m) most
Of the 3 inputs, launch speed (m/s) moves the range (m) most (16.3 across the range tested) and launch angle (degrees) moves it least (0.223).
| Input | Tested from | To | Range (m) at each end | Swing |
|---|---|---|---|---|
| Launch speed (m/s) | 18 | 22 | 33 to 49.3 | 16.3 (40%) |
| Gravity (m/s²) | 9 | 11 | 44.4 to 36.4 | 8.1 (20%) |
| Launch angle (degrees) | 41 | 50 | 40.4 to 40.2 | 0.223 (0.5%) |
Two variables at once
Range (m) by launch speed (m/s) and launch angle (degrees)
Across the grid the range (m) runs from 24.8 to 58.7. Moving launch speed (m/s) from 16 to 24 shifts it by 32.6 at the middle column, and moving launch angle (degrees) from 36 to 54 shifts it by 0 at the middle row, so launch speed (m/s) is the bigger lever here.
| Launch speed (m/s) \ Launch angle (degrees) | 36 | 45 | 54 |
|---|---|---|---|
| 16 | 24.8 | 26.1 | 24.8 |
| 18 | 31.4 | 33 | 31.4 |
| 20 | 38.8 | 40.8 | 38.8 |
| 22 | 46.9 | 49.3 | 46.9 |
| 24 | 55.8 | 58.7 | 55.8 |
The highlighted cell is the worked example: 40.8.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Launch speed (m/s) | 20 | Enter the launch speed (m/s) used in this calculation. |
| Launch angle (degrees) | 45 | Enter the launch angle (degrees) used in this calculation. |
| Gravity (m/s²) | 9.8 | Enter the gravity (m/s²) used in this calculation. |
| Range (m) | 40.8 | |
| Maximum height (m) | 10.2 | |
| Flight time (s) | 2.9 | |
Inputs, definitions and assumptions
Launch speed (m/s)
Enter the launch speed (m/s) used in this calculation. The prefilled worked-example value is 20.
Launch angle (degrees)
Enter the launch angle (degrees) used in this calculation. The prefilled worked-example value is 45.
Gravity (m/s²)
Enter the gravity (m/s²) used in this calculation. The prefilled worked-example value is 9.8.
How to use this calculator
- 1Verify the inputs. Gather launch speed (m/s), launch angle (degrees) and gravity (m/s²) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the range (m) at 40.8. Store your own version of it as Scenario A.
- 3Test one change. Start with launch speed (m/s), the input with the biggest effect here: moving launch speed (m/s) from 18 to 22 takes the range (m) from 33 to 49.3, a swing of 40% of the worked-example figure.
- 4Check the extremes. At half the example launch speed (m/s) (10) the range (m) is 10.2; at double (40) it is 163.1.
People also ask
Frequently asked questions
How do you calculate projectile motion?
Range = v² × sin 2θ ÷ g; maximum height = v² × sin² θ ÷ (2g); flight time = 2v × sin θ ÷ g, on level ground with no air resistance. At the worked-example inputs the range (m) is 40.8.
What does the projectile motion result mean?
Predict where a thrown or launched object lands on level ground. At the worked-example inputs the range (m) is 40.8. It rises with launch speed (m/s) and falls as gravity (m/s²) and launch angle (degrees) increase.
How much does launch speed (m/s) change the range (m)?
Holding every other input at the worked-example value, moving launch speed (m/s) from 16 to 24 moves the range (m) from 26.1 to 58.7, a spread of 32.6.
What are the limits of this projectile motion calculator?
These are textbook formulas for ideal conditions: no air resistance, no friction, gravity of 9.81 meters per second squared and gases that behave ideally. Real results differ, so treat them as first estimates and use the units the formulas expect (meters, kilograms, seconds). The tables on this page test launch speed (m/s) only from 16 to 24; a value outside that range is not tabulated here.
Which input moves the range (m) most in the projectile motion calculator?
Ranked by how far each moves the range (m) across the range tested: launch speed (m/s) (16.3, 40%), gravity (m/s²) (8.1, 20%) and launch angle (degrees) (0.223, 0.5%).
If I double launch speed (m/s) in the projectile motion calculator, does the range (m) double?
Doubling it from 20 to 40 takes the range (m) from 40.8 to 163.1, which is 4.00 times the worked-example figure. So the result grows faster than the input does. Halving it to 10 gives 10.2.
How much does launch angle (degrees) matter in the projectile motion calculator?
The worked example uses 45. Holding every other input at its worked-example value, moving launch angle (degrees) from 41 to 50 takes the range (m) from 40.4 to 40.2, a swing of 0.5% of the worked-example figure.
How much does gravity (m/s²) matter in the projectile motion calculator?
The worked example uses 9.8. With the other inputs left at the worked example, moving gravity (m/s²) from 9 to 11 takes the range (m) from 44.4 to 36.4, a swing of 20% of the worked-example figure.
Which inputs change the maximum height (m) in the projectile motion calculator?
At the worked-example inputs it is 10.2. Launch speed (m/s) takes it from 8.3 to 12.3, launch angle (degrees) takes it from 8.8 to 12 and gravity (m/s²) takes it from 11.1 to 9.1.
Which inputs change the flight time (s) in the projectile motion calculator?
At the worked-example inputs it is 2.9. Launch speed (m/s) takes it from 2.6 to 3.2, launch angle (degrees) takes it from 2.7 to 3.1 and gravity (m/s²) takes it from 3.1 to 2.6.
Is gravity the same everywhere on Earth?
Not quite. The standard value is 9.81 meters per second squared, but it runs from about 9.78 at the equator to about 9.83 at the poles, and it falls slightly with altitude. For everyday work 9.81 is plenty.
Sources and evidence
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