m/s
Please enter a valid initial velocity.
Speed at the moment of launch
degrees
Enter angle between 0 and 90.
Angle above horizontal. 45° = max range
m
Enter a valid height.
Height above landing surface. Leave 0 for ground level
Select gravity for different environments
Time of Flight
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Sources & Methodology
Formulas verified against HyperPhysics, Khan Academy Physics, and standard university-level classical mechanics references.
HyperPhysics — Trajectory of a Projectile
Georgia State University's authoritative physics reference for projectile motion equations, derivations, and worked examples used in this calculator.
Khan Academy — 2D Projectile Motion
Curriculum reference for projectile motion fundamentals, time of flight derivation, and the independence of horizontal and vertical motion.
NASA — Planetary Fact Sheet
Source for gravitational acceleration values on Earth (9.81), Moon (1.62), Mars (3.72), Venus (8.87), and Jupiter (24.79) used in the gravity selector.
Formulas used: Time of flight (ground level): T = 2v₀sinθ / g. With height h: T = [v₀sinθ + √((v₀sinθ)² + 2gh)] / g. Max height: H = h + (v₀sinθ)² / (2g). Range: R = v₀cosθ × T. Final speed: v_f = √(v₀² + 2gh) for h-launch, else = v₀.
⏱ Last reviewed: March 2026
How to Calculate Time of Flight for Projectile Motion
Projectile motion describes the curved path of an object launched at an angle under the influence of gravity, with no horizontal force acting after launch (ignoring air resistance). The motion can be split into two independent components: horizontal (constant velocity) and vertical (constant acceleration due to gravity).
Time of Flight Formula — Launched from Ground Level
T = (2 × v₀ × sin θ) / g
T = Total time of flight (seconds)
v₀ = Initial velocity (m/s)
θ = Launch angle above horizontal (degrees)
g = Gravitational acceleration (9.81 m/s² on Earth)
Example: v₀ = 20 m/s, θ = 45°:
T = (2 × 20 × sin 45°) / 9.81 = (2 × 20 × 0.7071) / 9.81 = 2.88 seconds
v₀ = Initial velocity (m/s)
θ = Launch angle above horizontal (degrees)
g = Gravitational acceleration (9.81 m/s² on Earth)
Example: v₀ = 20 m/s, θ = 45°:
T = (2 × 20 × sin 45°) / 9.81 = (2 × 20 × 0.7071) / 9.81 = 2.88 seconds
Maximum Height and Range Formulas
H = (v₀ × sin θ)² / (2g) R = v₀² × sin(2θ) / g
H = Maximum height (m) — reached at T/2
R = Horizontal range (m) — total distance traveled
Continuing the example (v₀ = 20 m/s, θ = 45°):
H = (20 × 0.7071)² / (2 × 9.81) = 200 / 19.62 = 10.19 m
R = 20² × sin(90°) / 9.81 = 400 / 9.81 = 40.77 m
R = Horizontal range (m) — total distance traveled
Continuing the example (v₀ = 20 m/s, θ = 45°):
H = (20 × 0.7071)² / (2 × 9.81) = 200 / 19.62 = 10.19 m
R = 20² × sin(90°) / 9.81 = 400 / 9.81 = 40.77 m
Projectile Motion at Different Launch Angles
The launch angle determines how the initial velocity is split between vertical and horizontal components. Different angles produce dramatically different trajectories:
| Angle (θ) | Flight Time (20 m/s) | Max Height | Range | Notes |
|---|---|---|---|---|
| 15° | 1.05 s | 1.36 m | 20.36 m | Low, fast arc |
| 30° | 2.04 s | 5.10 m | 35.35 m | Shallow trajectory |
| 45° | 2.88 s | 10.19 m | 40.77 m | Maximum range |
| 60° | 3.53 s | 15.29 m | 35.35 m | Same range as 30° |
| 75° | 3.94 s | 19.04 m | 20.36 m | Same range as 15° |
| 90° | 4.08 s | 20.39 m | 0 m | Straight up and down |
💡 Key insight: Complementary angles (e.g., 30° and 60°, or 15° and 75°) produce the same horizontal range but different flight times and maximum heights. The 45° angle is unique — it is the only angle that maximises horizontal range on level ground.
Projectile Motion on Other Planets
Gravity is the only factor that changes between planets — the initial velocity and launch angle stay the same. Lower gravity means longer flight time and greater range for the same launch conditions. A ball thrown at 20 m/s at 45° on the Moon would travel 252 m (vs 40.77 m on Earth) and stay airborne for 17.45 seconds.
| Body | g (m/s²) | Flight Time (20 m/s, 45°) | Range |
|---|---|---|---|
| Moon | 1.62 | 17.45 s | 247.2 m |
| Mars | 3.72 | 7.59 s | 107.6 m |
| Venus | 8.87 | 3.18 s | 45.1 m |
| Earth | 9.81 | 2.88 s | 40.8 m |
| Jupiter | 24.79 | 1.14 s | 16.1 m |
Real-World Applications of Projectile Motion
- Sports science: Basketball free throws, long jump trajectories, and football punts all follow projectile paths. Coaches use time of flight to analyze arc height and optimal release angles.
- Artillery and ballistics: The fundamental equations of projectile motion were historically developed to predict cannon range. Modern targeting systems still use these same principles, augmented for air resistance.
- Irrigation systems: Agricultural sprinklers are designed using projectile formulas to calculate the optimal nozzle angle and pressure to achieve maximum water distribution range.
- Stunt design: Film stunt coordinators and engineers calculate vehicle jump trajectories using time of flight formulas to ensure safe landing zone placement.
- Forensic ballistics: Investigators reconstruct shooting incidents by working backwards from impact location, height difference, and velocity to determine launch angle and position.
Why Air Resistance Is Ignored in These Calculations
The standard projectile motion equations assume a vacuum — no air resistance. In reality, air drag reduces both range and flight time, with the effect increasing at higher velocities and for less aerodynamic objects. For most physics problems, short-range ball throws, and educational purposes, ignoring air resistance gives results accurate enough to be practically useful. Professional applications (artillery, aerospace) use computational fluid dynamics (CFD) models that account for drag, wind, and the Magnus effect.
Frequently Asked Questions
The time of flight formula for a projectile launched from ground level is T = (2 × v₀ × sin θ) / g. For launch from height h, use T = [v₀sinθ + √((v₀sinθ)² + 2gh)] / g. Here v₀ is initial velocity (m/s), θ is launch angle, g is gravity (9.81 m/s² on Earth), and h is launch height above landing.
Use T = (2 × v₀ × sin θ) / g. A ball launched at 20 m/s at 45°: T = (2 × 20 × 0.707) / 9.81 = 2.88 seconds. The projectile spends exactly half the time (1.44 s) going up and the other half coming down, reaching maximum height at the midpoint of its flight.
A launch angle of 45° gives maximum range on flat ground. At 45°, sin(2θ) = sin(90°) = 1, which maximises R = v₀² × sin(2θ) / g. Complementary angles like 30° and 60°, or 20° and 70°, produce identical range but different trajectories — the shallower angle arrives faster and lower, the steeper angle arrives higher and slower.
Maximum height H = (v₀ × sin θ)² / (2g). If launched from height h above ground, H = h + (v₀ × sin θ)² / (2g). For v₀ = 20 m/s at 45°: H = (20 × 0.707)² / (2 × 9.81) = 200 / 19.62 = 10.19 m. Maximum height always occurs at exactly half the total flight time.
Launching from a height above the landing zone always increases total flight time. The projectile has to fall the extra height h in addition to its normal arc. The formula becomes T = [v₀sinθ + √((v₀sinθ)² + 2gh)] / g. For example, throwing a ball at 20 m/s at 45° from a 10 m cliff gives T = [14.14 + √(200 + 196.2)] / 9.81 = 4.20 s vs 2.88 s from ground level.
Range R = v₀² × sin(2θ) / g for ground-level launch, or R = v₀cosθ × T when launching from a height (where T is the full flight time from the height formula). For 20 m/s at 45°: R = 400 × sin(90°) / 9.81 = 40.77 m. Horizontal velocity is constant throughout flight because there is no horizontal force (ignoring air resistance).
Use g = 1.62 m/s² instead of 9.81. T = (2 × v₀ × sinθ) / 1.62. For 20 m/s at 45°: T = 28.28 / 1.62 = 17.45 s. Range = 400 × sin(90°) / 1.62 = 246.9 m. The Moon's lower gravity means the same throw travels about 6 times further and stays airborne over 6 times longer than on Earth. This is why astronauts can throw objects very far on the lunar surface.
Time to maximum height = v₀ × sinθ / g. This is exactly half the total flight time for a ground-level launch. For v₀ = 20 m/s at 45°: t_max = 20 × 0.707 / 9.81 = 1.44 s, which is T/2 = 2.88/2 = 1.44 s. The projectile decelerates vertically at rate g going up, and accelerates at the same rate g coming down, so each leg takes exactly the same time.
Yes, significantly for real-world objects. Air drag reduces both range and flight time. It also shifts the optimal angle for maximum range below 45° (typically 30°–40° depending on the object). For a baseball pitched at 90 mph, ignoring drag overestimates range by over 50%. This calculator uses the ideal (no air resistance) model, which is accurate for slow speeds and dense objects but increasingly approximate for faster projectiles in real air.
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