Calculate time of flight (T = [v₀·sinθ + √((v₀·sinθ)² + 2gh₀)] / g), hang time, free fall duration (t = √(2h/g)), ascent/descent intervals, and sensor TOF (LiDAR, Ultrasound, Mass Spec) with the free in-browser Time of Flight Calculator.
Ascent: 2.16 s • Descent: 2.16 s • Apex Height: 22.94 m • Range: 91.74 m
Time to reach apex
Time from apex to ground
Max vertical altitude
Ground displacement
Final landing velocity
High-precision units
In physics and engineering, Time of Flight (TOF) represents the total duration required for an object, particle, or wave packet to transit from an origin point to a destination target. In classical ballistics and mechanics, time of flight is dictated entirely by vertical initial velocity, gravitational acceleration, and elevation differentials:
Hang time depends exclusively on the vertical vector component (\(v_{0y} = v_0 \sin\theta\)). Horizontal speed has zero effect on how long an object remains airborne.
Under Galileo's principle of equivalence, all objects experience identical gravitational acceleration (\(g = 9.81\text{ m/s}^2\)) in a vacuum, irrespective of mass or composition.
Launching from an elevated platform (\(h_0 > 0\)) creates an asymmetric trajectory where descent duration strictly exceeds ascent time (\(t_{\text{descent}} > t_{\text{ascent}}\)).
The governing mathematical equations used by our Time of Flight Calculator:
| Physical Motion Scenario | Time of Flight Formula (\(T\)) | Key Governing Conditions |
|---|---|---|
| Flat Ground 2D Projectile | $$T = \frac{2v_0\sin\theta}{g}$$ | Symmetric ascent & descent (\(t_{\text{up}} = t_{\text{down}}\)) |
| Elevated Cliff Launch (\(h_0 > 0\)) | $$T = \frac{v_0\sin\theta + \sqrt{(v_0\sin\theta)^2 + 2gh_0}}{g}$$ | Quadratic solution for ground impact (\(y(t) = 0\)) |
| Vertical Launch Straight Up (\(\theta = 90^\circ\)) | $$T = \frac{2v_0}{g}$$ | Zero horizontal displacement (\(R = 0\)) |
| Free Fall from Rest (\(v_0 = 0\)) | $$t_{\text{fall}} = \sqrt{\frac{2h}{g}}$$ | Pure vertical drop under gravitational acceleration |
| Sensor / Wave Round-Trip (LiDAR/Sonar) | $$\tau = \frac{2d}{c}$$ | Round-trip echo timing at wave speed \(c\) |
Choose 2D Projectile, Vertical Launch, Free Fall Drop, or Sensor Time of Flight.
Enter launch speed (\(v_0\)) and adjust the launch angle slider (\(0^\circ\) to \(90^\circ\)).
Optionally set a cliff height offset (\(h_0\)) and select Earth, Moon, or Mars gravitational acceleration.
Inspect total hang time, ascent vs. descent duration breakdown, apex altitude, and complete step-by-step KaTeX derivation.
Why flight phases are perfectly symmetric on flat ground but asymmetric from elevated cliffs:
Gravity decelerates the object on the way up at \(-g\) and accelerates it downward at \(+g\). Consequently, \(t_{\text{ascent}} = t_{\text{descent}} = \frac{v_0 \sin\theta}{g}\).
The projectile must fall through both the apex height and the extra cliff elevation, making descent time strictly longer: \(t_{\text{descent}} = \sqrt{\frac{2(H_{\text{max}})}{g}} > t_{\text{ascent}}\).
How elite athletes optimize hang time across competitive disciplines:
Punters target high launch angles (55°–60°) allowing coverage teams to reach the returner before the catch
Michael Jordan's legendary 48-inch vertical produced an airtime of exactly \(T = 2\sqrt{2 \times 1.22 / 9.81} = 0.998\text{ seconds}\)
Maximizing springboard takeoff velocity to complete multiple twists and flips before landing
Supports 2D projectiles, vertical launches, free fall drops, horizontal projectiles, and optical/acoustic sensor TOF.
Isolates the exact time taken to reach apex peak versus duration spent in gravitational descent.
Interactive dual-color timeline illustrating the proportion of flight spent climbing versus descending.
One-click gravitational acceleration toggles for Earth (\(9.81\)), Moon (\(1.62\)), Mars (\(3.72\)), and Jupiter (\(24.79\text{ m/s}^2\)).
Computes one-way and round-trip transit timing for LiDAR laser pulses, radar antennas, and ultrasonic rangefinders.
Effortlessly inputs and outputs in meters, feet, kilometers, m/s, km/h, mph, seconds, and milliseconds.
How modern technology leverages nanosecond and microsecond time-of-flight measurements:
LiDAR sensors pulse laser photons and measure round-trip time (\(\tau\)) with picosecond accuracy. Measuring \(\tau = 667\text{ nanoseconds}\) determines target distance at \(d = c\tau/2 = \mathbf{100.0\text{ meters}}\).
Ions with identical kinetic energy (\(E_k = zeV\)) travel through a vacuum tube of length \(L\). Flight time scales with mass-to-charge ratio: \(t = L\sqrt{\frac{m}{2zeV}}\), resolving molecular masses to \(0.001\text{ Da}\).
Solves asymmetric vertical landing equations (\(h_0 + v_{0y}t - 0.5gt^2 = 0\)) instantly without manual quadratic formula calculations.
Bridges mechanical ballistics with optical LiDAR and acoustic sonar transit time in a single unified interface.
How local gravitational acceleration directly dictates hang time for a \(v_0 = 30\text{ m/s}\) at \(45^\circ\) projectile (\(v_{0y} = 21.21\text{ m/s}\)):
6.05x longer hang time than Earth
2.63x longer hang time than Earth
Standard 1.00x gravity baseline
0.40x rapid gravitational pull-down
Why objects falling from high altitudes exceed vacuum free fall calculations:
In a vacuum, dropping from \(4,000\text{ meters}\) requires \(t = \sqrt{2 \times 4000 / 9.81} = \mathbf{28.56\text{ seconds}}\). However, in Earth's atmosphere, a human skydiver reaches a terminal velocity of approximately \(v_t \approx 54\text{ m/s}\) (\(194\text{ km/h}\)), extending the actual free fall duration to approximately \(60\text{ seconds}\).
How diagnostic sonography maps human anatomy via acoustic transit times:
Ultrasound transducers emit high-frequency sound pulses and measure the echo return time (\(\tau\)). Given the average speed of sound in human soft tissue (\(c_{\text{tissue}} = 1,540\text{ m/s}\)), every \(13\text{ microseconds}\) of round-trip time corresponds to exactly \(1\text{ centimeter}\) of anatomical depth (\(d = c\tau/2\)).
How picosecond photon coincidence timing elevates cancer imaging resolution:
In TOF-PET oncology scans, positron annihilation emits two opposing \(511\text{ keV}\) gamma photons at the speed of light. Measuring the arrival time difference (\(\Delta t\)) between detector pairs localizes the tumor lesion along the Line of Response (LOR):
A timing resolution of \(300\text{ picoseconds}\) confines lesion positioning to within \(4.5\text{ cm}\), dramatically increasing image signal-to-noise ratio and detecting millimeter-scale metastases.
How multibeam echo sounders (MBES) map deep ocean seabed topography:
Acoustic sonar pulses travel to the sea floor and reflect back: \(d = (v_{\text{water}} \cdot \tau) / 2\). In the \(11,000\text{-meter}\) Mariana Trench, two-way acoustic transit takes approximately \(14.6\text{ seconds}\).
Ocean salinity, temperature, and hydrostatic pressure alter sound speed from \(1,450\text{ to }1,550\text{ m/s}\), requiring continuous SVP calibration for sub-meter depth precision.
Calculating interplanetary flight durations between planetary orbits:
Interplanetary spacecraft travel along elliptical transfer orbits governed by Kepler's Third Law. The flight time for a minimum-energy Hohmann transfer from Earth to Mars is half the orbital period:
Where \(a = (r_{\text{Earth}} + r_{\text{Mars}})/2 = 1.262\text{ AU}\) is the semi-major axis of the elliptical heliocentric trajectory.
Comprehensive answers to common questions about time of flight formulas, hang time, free fall duration, and sensor TOF applications.
