Projectile Motion Calculator
A ball launched at 20 m/s and 45° from the ground lands 40.7886 m away, rising to 10.2 m and staying in the air 2.88 s (no air resistance).
Above the horizontal, from −90° to 90°.
40.7886485191171 m
Full answer, then cut to fewer decimals (not rounded):
- 40.7886485191171 m
- 40.788648519117 m
- 40.78864851911 m
- 40.7886485191 m
- 40.788648519 m
- 40.78864851 m
- 40.7886485 m
- 40.788648 m
- 40.78864 m
- 40.7886 m
- 40.788 m
- 40.78 m
- 40.7 m
- Time of flight T
- 2.884192996 s
- Maximum height H
- 10.19716213 m
- Time to the top
- 1.442096498 s
- Horizontal velocity vₓ
- 14.14213562 m/s
- Initial vertical velocity v_y
- 14.14213562 m/s
- Impact speed
- 20 m/s
- Impact angle below horizontal
- 45 °
Step-by-step solution
- vₓ = v₀ cos θ = 20 × cos 45° = 14.14213562 m/s; v_y = v₀ sin θ = 14.14213562 m/s
- T = 2 v_y ÷ g = 2 × 14.14213562 ÷ 9.80665 = 2.884192996 s
- R = vₓ × T = 14.14213562 × 2.884192996 = 40.78864852 m
- H = h₀ + v_y² ÷ 2g = 0 + 14.14213562² ÷ (2 × 9.80665) = 10.19716213 m
Projectile Motion Formula
R = v₀ cos θ × T, H = h₀ + (v₀ sin θ)²/2g
T = (v₀ sin θ + √((v₀ sin θ)² + 2gh₀))/g; on level ground R = v₀² sin 2θ / g
Worked Example
A ball kicked at 20 m/s and 45° from ground level (g = 9.80665 m/s²):
- vₓ = v₀ cos θ = 20 × cos 45° = 14.14213562 m/s; v_y = v₀ sin θ = 14.14213562 m/s
- T = 2 v_y ÷ g = 2 × 14.14213562 ÷ 9.80665 = 2.884192996 s
- R = vₓ × T = 14.14213562 × 2.884192996 = 40.78864852 m
- H = h₀ + v_y² ÷ 2g = 0 + 14.14213562² ÷ (2 × 9.80665) = 10.19716213 m
Maximum height H: 10.19716213 m · Time to the top: 1.442096498 s · Horizontal velocity vₓ: 14.14213562 m/s · Initial vertical velocity v_y: 14.14213562 m/s · Impact speed: 20 m/s · Impact angle below horizontal: 45 °
Horizontal and vertical motion are independent
The horizontal velocity v₀ cos θ stays constant while gravity changes only the vertical part. Solving each direction separately, then joining them through the shared time T, is the method behind every result here.
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Frequently asked questions
Which angle gives the longest range?
On level ground, 45°. Launched from a height, the best angle is a little lower; with air resistance it is lower still, around 35–40° for a thrown ball.
Why do 30° and 60° give the same range?
On level ground range depends on sin 2θ, and sin 60° = sin 120°. Complementary angles land in the same place, but the steeper one flies higher and longer.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Projectile Motion Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/projectile-motion-calculator/
IEEE
Scientific Converters, "Projectile Motion Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/projectile-motion-calculator/