Velocity Calculator

How to use this calculator

Select what you want to find: Velocity, Displacement, Time, or Final Velocity (using kinematics). For the basic v = d/t mode, enter any two of velocity, displacement, and time. For the kinematic mode, enter initial velocity, acceleration, and time to find final velocity.

Use the unit selectors to work in metres, kilometres, miles, feet, or light-years for distance, and seconds, minutes, or hours for time. The calculator converts automatically.

Velocity vs speed: the key distinction

Velocity and speed are related but not the same.

Speed is a scalar quantity: it has magnitude only. A car travelling at 60 km/h has a speed of 60 km/h regardless of direction.

Velocity is a vector quantity: it has both magnitude and direction. A car travelling at 60 km/h northward has a velocity of 60 km/h north. A car going 60 km/h southward has the same speed but a completely different velocity.

This distinction matters when direction changes. A car completing a full lap of a circuit returns to its starting point. Total distance = full lap length. Total displacement = 0. Average speed = lap length / time. Average velocity = 0 (no net displacement).

In one dimension, velocity can be positive or negative depending on the chosen direction convention. In two or three dimensions, velocity is described by components: v = (vₓ, vᵧ) or (vₓ, vᵧ, v_z).

The velocity formula

For constant velocity (no acceleration), the relationship between velocity, displacement, and time is:

Where Δx is displacement (change in position) and Δt is elapsed time. Rearranging:

This is average velocity. Instantaneous velocity is the limit of Δx/Δt as Δt approaches zero: the derivative of position with respect to time.

On a position-time graph, instantaneous velocity at any point is the slope of the tangent to the curve at that point. For constant velocity, the curve is a straight line and slope = average velocity = instantaneous velocity everywhere.

Kinematic equations for velocity with acceleration

When velocity changes at a constant rate (constant acceleration), a set of four kinematic equations applies. These relate initial velocity (u), final velocity (v), acceleration (a), displacement (s), and time (t).

These equations assume constant acceleration. They apply to uniformly accelerating objects: free-fall (ignoring air resistance), a car braking at constant deceleration, or a rocket with constant thrust.

Velocity in two dimensions

In two dimensions, velocity is a vector with x and y components. If an object moves with speed v at angle θ from the horizontal:

The magnitude of the velocity vector (speed) and direction can be recovered from components:

This decomposition is essential for projectile motion. A ball thrown at 20 m/s at 45°:

• Horizontal velocity: vₓ = 20 cos 45° = 14.14 m/s (constant, no horizontal acceleration)
• Vertical velocity: vᵧ = 20 sin 45° = 14.14 m/s (decreases at 9.81 m/s² due to gravity)

The horizontal and vertical motions are independent and can be treated separately. This is the key insight of Galilean kinematics.

Velocity-time graphs

A velocity-time graph plots velocity on the y-axis and time on the x-axis. Several important quantities can be read directly from such a graph.

Slope = acceleration: The gradient of the line at any point gives the instantaneous acceleration. A flat horizontal line means constant velocity (zero acceleration). A steep upward slope means large positive acceleration.

Area = displacement: The area under the v-t curve between two times equals the displacement in that time interval. For a rectangular section (constant velocity), area = v × t. For a triangular section (uniform acceleration from rest), area = ½ × v_max × t.

Relative velocity

Velocity depends on the reference frame from which it is measured. Relative velocity is the velocity of one object as observed from another moving object.

For two objects A and B moving in the same direction:

For opposite directions, the relative speed is the sum of the individual speeds.

In Einstein’s special relativity, velocities do not simply add for speeds approaching the speed of light. The relativistic addition formula is:

At everyday speeds, v₁v₂/c² is essentially zero and the classical formula is accurate. At speeds above 10% of c, the correction becomes significant.

Escape velocity

Escape velocity is the minimum velocity needed for an object to escape a gravitational field without further propulsion. Derived by setting kinetic energy equal to gravitational potential energy:

Where G is the gravitational constant, M is the mass of the body, and r is the distance from its centre.

Body	Escape velocity
Earth	11.2 km/s
Moon	2.38 km/s
Mars	5.03 km/s
Jupiter	59.5 km/s
Sun	617.5 km/s
Neutron star (typical)	~100,000 km/s (~c/3)

The Moon’s low escape velocity explains why it has no significant atmosphere: gas molecules with typical thermal velocities can escape over geological time. Earth’s heavier molecules (N₂, O₂, CO₂) have thermal velocities well below escape velocity, so the atmosphere is retained. Lighter hydrogen and helium slowly leak away.

Velocity in circular motion

An object in circular motion at constant speed has a changing velocity. The speed is constant, but the direction continuously changes, meaning there is always a nonzero acceleration directed toward the centre (centripetal acceleration).

Where ω is angular velocity in radians per second and r is the radius of the circle. The centripetal acceleration required is:

For a car rounding a curve: if the car moves too fast, the required centripetal force (v²/r × m) exceeds the available friction force and the car slides outward. The maximum safe cornering speed is:

Where μ is the friction coefficient. A curve with radius 50 m and μ = 0.7 has a maximum safe speed of √(0.7 × 9.81 × 50) = 18.5 m/s = 67 km/h.

Velocity measurement methods

Doppler radar: Emits microwave pulses and measures the frequency shift of the reflected signal. The shift is proportional to the component of velocity along the radar beam. Used by traffic police, weather services, and sports (pitch speed, tennis serve speed).

GPS: Consecutive position fixes with known time intervals give average velocity. More precise GPS units also directly compute instantaneous velocity from Doppler shift of satellite signals.

Accelerometer integration: Integrating acceleration over time gives velocity change. Inertial navigation systems in aircraft and spacecraft use accelerometers to compute velocity from a known initial state.

Optical Doppler: Laser Doppler velocimetry fires two laser beams at a fluid. Particles in the flow scatter light with a frequency shift proportional to velocity. Used in research to measure flow fields without inserting physical probes.

Pitot tube: Measures the dynamic pressure of a fluid flow. Combined with static pressure, gives airspeed via Bernoulli’s equation. Standard in aircraft.

Notable velocities in science and nature

Phenomenon	Velocity
Plate tectonic movement	~3-10 cm/year
Human walking	~1.4 m/s
Cheetah (top speed)	~31 m/s
Sound in air (20°C)	343 m/s
Rifle bullet	~900 m/s
Earth orbital velocity	29,800 m/s
Parker Solar Probe (max)	~192,000 m/s
Speed of light (vacuum)	299,792,458 m/s

The orbital velocity formula gives Earth’s orbital speed from gravitational mechanics: v_orbit = √(GM_sun / r) = √(6.674×10⁻¹¹ × 2×10³⁰ / 1.496×10¹¹) = 29,800 m/s. This is the same result from the definition: Earth travels ~942 million km per year in roughly 365.25 days.