Projectile Motion Calculator - Range, Height, Time & Velocity Solver

Q: What is the formula for projectile motion range?

The range formula is R = v₀²sin(2θ)/g, where v₀ is initial speed, θ is launch angle, and g is gravitational acceleration (9.81 m/s² on Earth).

Q: Why does 45° give maximum range?

The range formula R = v₀²sin(2θ)/g is maximized when sin(2θ)=1, which occurs when 2θ=90°, giving θ=45° and R_max=v₀²/g.

Q: How do you find the maximum height of a projectile?

Maximum height H = (v₀sinθ)²/(2g), reached when vertical velocity equals zero at time t_peak = v₀sinθ/g.

Q: What is the time of flight formula?

For same-height launch and landing: T = 2v₀sinθ/g. For elevated launch from height h: T = [v₀sinθ + √((v₀sinθ)² + 2gh)] / g.

Shahid Ali

Lead Developer · CECOS University of IT and Emerging Sciences · ORCID: 0009-0008-5995-4438

Fact-Checked DOI Archived Last verified: August 2026

🚀 Projectile Motion Solver

Initial Speed v₀

Launch Angle θ

Init. Height h₀

Launch Angle 45°

★ Maximum range at 45°

🎬 Trajectory Visualizer

Speed: 1×

📈 Range vs Launch Angle

📍 Position at Time t

Time t (seconds)

Projectile Motion Equations — Complete Formula Reference

Projectile motion separates into two independent components: horizontal (constant velocity) and vertical (constant acceleration due to gravity).

R = v₀²sin(2θ)/g H = (v₀sinθ)²/(2g) T = 2v₀sinθ/g

Quantity	Formula	Notes
Horiz. velocity	`vₓ = v₀cosθ`	Constant throughout flight
Initial vert. velocity	`vy₀ = v₀sinθ`	At launch
Vert. velocity at t	`vy = v₀sinθ − gt`	Decreases with time
Horiz. position	`x = v₀cosθ · t`	Linear in time
Vert. position	`y = v₀sinθ · t − ½gt²`	Parabolic path
Time to peak	`t_peak = v₀sinθ / g`	Half of total flight time
Maximum height	`H = (v₀sinθ)² / (2g)`	At t = t_peak
Total flight time	`T = 2v₀sinθ / g`	Same-height launch only
Horizontal range	`R = v₀²sin(2θ) / g`	Same-height landing only
Speed at time t	`v = √(vₓ² + vy²)`	Pythagorean combination

How Projectile Motion Works — The Two-Component Method

The key principle: horizontal and vertical motions are completely independent. Solve each separately, then combine.

↔ Horizontal

No force acts horizontally
Velocity stays constant: vₓ = v₀cosθ
Distance: x = vₓ × t

↕ Vertical

Gravity pulls down at g = 9.81 m/s²
Velocity changes: vy = vy₀ − gt
Height: y = vy₀t − ½gt²

The range formula R = v₀²sin(2θ)/g is maximized when sin(2θ) = 1, i.e. when 2θ = 90°, giving θ = 45°. Maximum height is found using H = (v₀sinθ)²/(2g) — only the vertical velocity component contributes to height. Complementary angles (e.g. 30° and 60°) produce identical ranges but different heights and flight times.

Angle θ	Range (m)	Max Height (m)	Time (s)
15°	20.39	1.34	1.05
30°	35.35	5.10	2.04
45° ★	40.77	10.19	2.88
60°	35.35	15.29	3.53
75°	20.39	19.05	3.93
90°	0	20.39	4.08

Values for v₀ = 20 m/s on Earth (g = 9.81 m/s²)

Why 45° Gives Maximum Range — The Mathematics Explained

The range formula is R = v₀²sin(2θ)/g. To maximize R we must maximize sin(2θ). Since the maximum value of any sine is 1, and sin(2θ)=1 when 2θ=90°, the optimal angle is θ = 45°, giving R_max = v₀²/g.

Initial Speed v₀	R_max at 45° (m)	R_max at 45° (ft)
5 m/s	2.55	8.36
10 m/s	10.19	33.43
20 m/s	40.77	133.7
30 m/s	91.74	300.8
50 m/s	254.8	835.7
100 m/s	1,019	3,343

ℹ️ Note: with air resistance the optimal angle drops below 45° (typically 30–40° depending on the object's drag). This calculator assumes no air resistance.

Projectile Motion on the Moon and Other Planets

Gravity determines everything in projectile motion. On the Moon (g = 1.62 m/s²) the same kick sends a ball 6× farther. Use the planet selector above to compare.

Body	g (m/s²)	Range (m)	Max H (m)	Time (s)
🌍 Earth	9.81	40.77	10.19	2.88
🌕 Moon	1.62	246.9	61.73	17.42
🔴 Mars	3.72	107.5	26.88	7.58
🟤 Jupiter	24.79	16.14	4.03	1.14
⚫ Mercury	3.70	108.1	27.03	7.62
🟡 Venus	8.87	45.07	11.27	3.18

Values for v₀ = 20 m/s, θ = 45°

🏌️ During Apollo 14 (1971), astronaut Alan Shepard hit a golf ball on the Moon. With g = 1.62 m/s², even a moderate swing sent the ball vast distances in the low gravity — he estimated it went "miles and miles."

Projectile Motion in Real Life — Sports, Engineering & Nature

Projectile motion appears everywhere — from a football kick to a water fountain arc. Here are five real applications with worked numbers.

⚽ Football / Soccer

A penalty kick at v₀ = 28 m/s, θ = 16° travels 11 m horizontally.
x = v₀cosθ × t → t = 11 / (28 × cos16°) = 11 / 26.93 = 0.409 s
y = 28×sin16°×0.409 − ½×9.81×0.409² = 3.14 − 0.82 = 2.32 m — clears a 2.44 m crossbar only slightly. Launch angle matters enormously.

🏀 Basketball Free Throw

A free throw launched at v₀ = 7 m/s, θ = 51° from h₀ = 2 m must reach a basket at x = 4.6 m, y = 3.05 m.
t = 4.6 / (7 × cos51°) = 4.6 / 4.404 = 1.044 s
y = 2 + 7×sin51°×1.044 − ½×9.81×1.044² = 2 + 5.676 − 5.351 = 2.32 m — just below rim. The classic 51° "optimal" angle is validated by this calculation.

🏃 Long Jump

An athlete leaves the board at v₀ = 9.5 m/s, θ = 22°.
R = 9.5² × sin(44°) / 9.81 = 90.25 × 0.6947 / 9.81 = 6.39 m
World-class long jumpers achieve 8+ m because their takeoff speed exceeds 10 m/s. Every extra m/s at launch adds roughly 1.5 m to the range.

💧 Water Fountain Arc

Decorative fountains are designed using projectile equations to hit a target landing point. A nozzle angled at 60° with v₀ = 4 m/s:
R = 4² × sin(120°) / 9.81 = 16 × 0.866 / 9.81 = 1.41 m
H = (4×sin60°)² / (2×9.81) = (3.464)² / 19.62 = 0.612 m — engineers use these numbers to route water precisely into basins.

🪖 Ballistics

This calculator uses simplified physics without air resistance — the standard assumption in introductory physics. Real ballistics software includes drag coefficients, wind, the Coriolis effect, and spin-induced deflection (Magnus effect). For a rifle bullet at v₀ = 900 m/s and θ = 0.5°, vacuum range = 900² × sin(1°) / 9.81 = 1,413 m, but actual range is far shorter due to aerodynamic drag.

Common Mistakes in Projectile Motion Problems

These five errors account for the majority of wrong answers in projectile motion problems:

❌ Mistake 1 — Using v₀ directly in vertical equations instead of vy₀

Wrong: H = v₀² / (2g) → gives a height that's too large
Correct: H = (v₀sinθ)² / (2g) → only the vertical component lifts the projectile
For v₀=20, θ=30°: Wrong gives H=20.4 m, correct gives H=5.1 m

❌ Mistake 2 — Forgetting to separate horizontal and vertical components

Wrong: Applying kinematic equations to the full speed v₀ directly
Correct: Always split first — vₓ = v₀cosθ (horizontal) and vy₀ = v₀sinθ (vertical) — then apply kinematic equations to each direction independently

❌ Mistake 3 — Using g as positive in the vertical position equation

Wrong: y = vy₀t + ½gt² → projectile goes up forever
Correct: y = vy₀t − ½gt² → gravity decelerates upward motion (g is magnitude; the minus sign accounts for direction)

❌ Mistake 4 — Confusing total flight time with time to peak

Wrong: Using T = v₀sinθ/g as the total flight time
Correct: t_peak = v₀sinθ/g is time to reach maximum height; total flight T = 2t_peak (for same-height launch). Using t_peak gives half the correct range.

❌ Mistake 5 — Using R = v₀²sin(θ)/g instead of R = v₀²sin(2θ)/g

Wrong: R = v₀²sin(θ)/g → incorrect formula
Correct: R = v₀²sin(2θ)/g → the argument is 2θ, not θ. At 45°: sin(45°) = 0.707 vs sin(90°) = 1.0 — the wrong formula gives 29% less range.

Frequently Asked Questions

What is the formula for projectile motion range?

The range formula is R = v₀²sin(2θ)/g, where v₀ is initial speed, θ is launch angle, and g is gravitational acceleration (9.81 m/s² on Earth). This formula applies when the projectile lands at the same height as it was launched. For elevated launches use the full parametric equations with the quadratic time formula.

Why does 45° give maximum range in projectile motion?

The range formula R = v₀²sin(2θ)/g contains sin(2θ). The sine function reaches its maximum value of 1 when its argument equals 90°. So sin(2θ) is maximized when 2θ = 90°, meaning θ = 45°. At this angle the maximum range is R_max = v₀²/g. In reality, air resistance lowers the optimal angle to roughly 30–40°.

How do you find the maximum height of a projectile?

Maximum height is reached when the vertical velocity equals zero. Using vy = 0: H = (v₀sinθ)² / (2g). For example, with v₀ = 20 m/s and θ = 45°: vy₀ = 14.14 m/s, so H = 14.14² / (2 × 9.81) = 200/19.62 = 10.19 m. The time to reach this height is t_peak = v₀sinθ/g.

What is the time of flight formula?

For a projectile launched and landing at the same height: T = 2v₀sinθ/g. This equals exactly twice the time to reach maximum height. For an elevated launch from height h: T = [v₀sinθ + √((v₀sinθ)² + 2gh)] / g, which comes from solving the quadratic equation for when y(t) = 0.

What is ballistic motion?

Ballistic motion is the trajectory followed by an object acted on only by gravity after being launched — identical to projectile motion. The term "ballistic" comes from ballistics (the science of projectiles like bullets and shells). In this calculator, ballistic motion, projectile motion, and parabolic motion all refer to the same physical phenomenon.

How do you find the initial velocity of a projectile?

If you know the range and launch angle: v₀ = √(Rg / sin(2θ)). If you know the maximum height and angle: v₀ = √(2gH) / sinθ. If you know range and height: first find θ = atan(4H/R), then v₀ = √(gR / sin(2θ)). Enter any two known values in the calculator above and it solves for v₀ automatically.

Do complementary angles give the same range?

Yes. Because R = v₀²sin(2θ)/g, and sin(2θ) = sin(180° − 2θ), angles θ and (90° − θ) produce identical ranges. For example, 30° and 60° give the same range. However, 60° gives a higher maximum height and longer flight time. Only 45° is its own complement and gives the unique maximum range.

Does air resistance affect projectile motion?

Yes, significantly in reality. This calculator assumes no air resistance (vacuum conditions) for simplicity, which is the standard assumption in introductory physics. With air resistance, the optimal angle drops below 45°, the range is shorter, and the trajectory is asymmetric (steeper descent than ascent). Real ballistics software includes drag coefficients, wind, and spin effects.

Related Calculators

Data Sources & References

Projectile motion equations and trajectory: Halliday, D., Resnick, R. & Krane, K.S. (2002). Physics, 5th ed. Wiley. Chapter 4.
Range and maximum height derivations: Serway, R.A. & Jewett, J.W. (2014). Physics for Scientists and Engineers, 9th ed. Cengage. Chapter 4.3.
Standard acceleration of gravity g = 9.80665 m/s²: CODATA 2018, NIST SP 961. physics.nist.gov
Optimal launch angle 45° for maximum range on level ground: Halliday, D., Resnick, R. & Krane, K.S. (2002). Physics, 5th ed. Wiley. Chapter 4, Problem 33.

Shahid Ali — Lead Developer, CECOS University Academically Verified

🆔 ORCID: 0009-0008-5995-4438 · 🔭 DOI: 10.5281/zenodo.21796048 · 📊 Figshare Dataset · 📋 Editorial Standards

Calculator methods archived with DOI · Last verified: August 2026

Projectile Motion Calculator – Range, Height, Time & Velocity Solver

🎯 Projectile Motion Calculator