Projectile Range
Calculator
Results
- Horizontal range
- 40.788648
- Maximum height
- 10.197162
- Time of flight
- 2.884192
Physics results
| Horizontal range | 40.788648 |
| Maximum height | 10.197162 |
| Time of flight | 2.884192 |
formula-map diagram
- Horizontal range
- 40.788648
- Maximum height
- 10.197162
- Time of flight
- 2.884192
Physical relationship
Formula
R = v² × sin(2θ) ÷ g= 40.788648519117
Note
This result applies an idealized textbook equation to the numbers you entered; it ignores air resistance, material tolerances and other real-world losses.
More in Physics and engineering
See all →Frequently asked questions
What does the projectile range formula R = v²sin(2θ)/g calculate?+
It gives the horizontal distance a projectile travels before landing back at its original launch height, based on launch speed and launch angle, assuming no air resistance. It's a simplified case of projectile motion that only applies when launch and landing heights are equal.
Why does a 45-degree launch angle give the maximum range?+
Range depends on sin(2θ), which reaches its maximum value of 1 when 2θ = 90°, meaning θ = 45°; any angle above or below 45 degrees produces a smaller sine value and therefore a shorter range for the same launch speed. This is why 45 degrees is the textbook 'ideal' angle for maximum distance on level ground, though real-world factors like air resistance shift the optimal angle lower in practice.
Why do two different launch angles sometimes give the same range?+
Because sin(2θ) is symmetric around 45 degrees, complementary angles like 30° and 60°, or 20° and 70°, produce identical sine values and therefore identical ranges at the same speed. One of the pair is a flatter, faster trajectory and the other is a higher, slower-arcing one, but they cover the same horizontal distance.
Does this formula work if the projectile lands at a different height than it was launched from?+
No — this specific formula assumes launch and landing occur at the same height, which is accurate for a ball thrown and caught at the same level but not for one launched from a cliff or landing in a valley. Different, more complete kinematic equations are needed when launch and landing heights differ.
Why does this calculator ignore air resistance, and how much does that matter in practice?+
Ignoring air resistance keeps the math to a clean, closed-form equation, and it's a reasonable approximation for dense, compact, slow-moving projectiles over short distances (like a thrown ball). For fast, light, or long-range projectiles (a golf ball, an arrow, or anything moving at high speed over a long trajectory), air resistance measurably reduces real range below what this idealized formula predicts.