Equation for Velocity: Velocity Equals Displacement Divided by Time, With 12 Formulas for Different Motion Types
The equation for velocity is v = Δx / Δt. Velocity equals displacement divided by elapsed time. Physics uses 12 velocity formulas for average, instantaneous, final, circular, wave, relative, orbital, escape, and terminal velocity. This article defines each formula, lists units, and shows 7 worked examples.
What Is the Equation for Velocity?
The equation for velocity is v = Δx / Δt, where velocity (v) equals displacement (Δx) divided by elapsed time (Δt). According to OpenStax University Physics, average velocity equals the change in position between two points divided by the time needed to travel between them.
Velocity is a vector quantity. Vector quantities have magnitude and direction. Examples include displacement, velocity, acceleration, and force. A velocity of 15 m/s east has a magnitude of 15 m/s and an eastward direction.
The table below defines each symbol in the velocity equation.
| Symbol | Quantity | SI Unit | Definition |
|---|---|---|---|
| v | Velocity | m/s | Rate of change of position with direction |
| Δx | Displacement | m | Final position minus initial position (x₂ – x₁) |
| Δt | Elapsed time | s | Final time minus initial time (t₂ – t₁) |
Velocity is negative when motion opposes the chosen positive direction. OpenStax describes a trip with a net displacement of -0.75 km in 58 minutes. The average velocity for that trip is -0.013 km/min.
What Are the 12 Velocity Formulas?
Physics has 12 main velocity formulas. The table below lists each formula by velocity type, equation, and condition of use.
| No. | Velocity Type | Equation | Condition of Use |
|---|---|---|---|
| 1 | Average velocity | vavg = Δx / Δt | Any motion over a time interval |
| 2 | Instantaneous velocity | v = dx / dt | Velocity at a single moment |
| 3 | Final velocity with known time | v = v₀ + at | Constant acceleration |
| 4 | Final velocity with unknown time | v² = v₀² + 2aΔx | Constant acceleration |
| 5 | Average velocity with constant acceleration | vavg = (v₀ + v) / 2 | Constant acceleration |
| 6 | Free-fall velocity | v = v₀ – gt | Vertical motion, upward positive |
| 7 | Circular motion velocity | v = 2πr / T | Uniform circular motion |
| 8 | Wave velocity | v = fλ | Periodic waves |
| 9 | Relative velocity | vAB = vA – vB | Two moving objects in one reference frame |
| 10 | Orbital velocity | v = √(GM / r) | Circular orbit |
| 11 | Escape velocity | v = √(2GM / r) | Leaving a gravitational field |
| 12 | Terminal velocity | vt = √(2mg / (ρ Cd A)) | Fall through a fluid |
The list below defines each symbol used in the table.
- v: velocity (m/s)
- v₀: initial velocity (m/s)
- vavg: average velocity (m/s)
- vA, vB: velocities of objects A and B (m/s)
- vAB: velocity of object A relative to object B (m/s)
- a: acceleration (m/s²)
- t: time (s)
- Δx: displacement (m)
- g: gravitational acceleration, 9.8 m/s² near Earth's surface
- r: radius (m)
- T: period of one revolution (s)
- f: frequency (Hz)
- λ: wavelength (m)
- G: gravitational constant, 6.674 × 10⁻¹¹ N·m²/kg²
- M: mass of the central body (kg)
- m: mass of the falling object (kg)
- ρ: fluid density (kg/m³)
- Cd: drag coefficient (dimensionless)
- A: projected area (m²)
What Is the Average Velocity Equation?
The average velocity equation is vavg = Δx / Δt = (x₂ – x₁) / (t₂ – t₁). It gives the constant velocity that produces the same displacement in the same time interval, according to ComPADRE NEXUS/Physics. Average velocity equals instantaneous velocity if velocity stays constant, as stated in OpenStax Physics.
What Is the Instantaneous Velocity Equation?
The instantaneous velocity equation is v = dx / dt, the limit of Δx / Δt as Δt approaches zero. Instantaneous velocity describes motion at one moment. The slope of a position-time graph at a given time equals the instantaneous velocity at that time, per the OpenStax chapter summary.
What Are the Constant Acceleration Velocity Equations?
Constant acceleration produces 4 velocity equations, and the most used equation is v = v₀ + at. The OpenStax key equations list all 4 forms:
- v = v₀ + at: finds final velocity when time is known
- v² = v₀² + 2aΔx: finds final velocity when time is unknown
- vavg = (v₀ + v) / 2: finds average velocity from initial and final velocity
- v = v₀ – gt: finds free-fall velocity with upward as the positive direction
Galileo's inclined plane experiments showed that distance from rest is directly proportional to the square of elapsed time under uniform acceleration, as described in a Honolulu Community College physics lab. This result agrees with x = x₀ + v₀t + ½at².
What Is the Circular Motion Velocity Equation?
The circular motion velocity equation is v = 2πr / T, where r is the radius and T is the time for one revolution. Tangential velocity also equals ωr, where ω is angular velocity. Angular velocity follows ω = Δθ / Δt and is measured in radians per second (rad/s). Uniform circular motion has constant speed and a changing velocity direction.
What Is the Wave Velocity Equation?
The wave velocity equation is v = fλ, where f is frequency and λ is wavelength. Sound in air at 20 °C travels at about 343 m/s. Light in a vacuum travels at 299,792,458 m/s.
What Is the Relative Velocity Equation?
The relative velocity equation is vAB = vA – vB, where vAB is the velocity of object A relative to object B. Two cars moving east at 30 m/s and 20 m/s have a relative velocity of 10 m/s. If car B moves west at 20 m/s, the relative velocity is 30 – (-20) = 50 m/s.
What Are the Orbital and Escape Velocity Equations?
Orbital velocity is v = √(GM / r), and escape velocity is v = √(2GM / r). Escape velocity equals √2 times orbital velocity at the same radius. A University of Connecticut handout reports an Earth escape velocity of about 11.2 km/s and a low Earth orbital velocity of about 7.9 km/s.
What Is the Terminal Velocity Equation?
The terminal velocity equation is vt = √(2mg / (ρ Cd A)). Terminal velocity is the constant velocity reached when drag force equals weight. It depends on mass, drag coefficient, projected area, and fluid density. Fluids include air, water, and oil.
How Do You Calculate Velocity Step by Step?
To calculate velocity, divide displacement by elapsed time. Follow these 6 steps:
- Identify the initial and final positions.
- Subtract the initial position from the final position to get displacement.
- Record the elapsed time between the two positions.
- Convert all values to SI units, meters and seconds.
- Divide displacement by elapsed time.
- State the result with a unit and a direction.
What Are Examples of Velocity Calculations?
The table below shows 7 worked examples. Each row lists the velocity type, known values, calculation, and result.
| No. | Velocity Type | Known Values | Calculation | Result |
|---|---|---|---|---|
| 1 | Average velocity | x₁ = 40 m, x₂ = 190 m, Δt = 10 s | (190 – 40) / 10 | 15 m/s |
| 2 | Final velocity with known time | v₀ = 5 m/s, a = 2 m/s², t = 6 s | 5 + (2 × 6) | 17 m/s |
| 3 | Final velocity with unknown time | v₀ = 0 m/s, a = 3 m/s², Δx = 24 m | √(0² + 2 × 3 × 24) = √144 | 12 m/s |
| 4 | Free-fall velocity | v₀ = 0 m/s, g = 9.8 m/s², t = 3 s | 0 – (9.8 × 3) | -29.4 m/s (downward) |
| 5 | Circular motion velocity | r = 0.5 m, T = 2 s | (2 × π × 0.5) / 2 | 1.57 m/s |
| 6 | Wave velocity | f = 440 Hz, λ = 0.78 m | 440 × 0.78 | 343.2 m/s |
| 7 | Escape velocity | G = 6.674 × 10⁻¹¹ N·m²/kg², M = 5.972 × 10²⁴ kg, r = 6.371 × 10⁶ m | √(2 × G × M / r) | 11,186 m/s (11.2 km/s) |
What Is the Difference Between Speed and Velocity?
Speed is a scalar quantity based on distance, and velocity is a vector quantity based on displacement. OpenStax defines average speed as total distance divided by elapsed time. The table below compares the two quantities by 6 properties.
| Property | Speed | Velocity |
|---|---|---|
| Quantity type | Scalar | Vector |
| Equation | Distance / time | Displacement / time |
| Direction | Not included | Included |
| Possible values | Zero or positive | Positive, negative, or zero |
| SI unit | m/s | m/s |
| Value for a full lap | Positive | Zero |
A runner who completes a 400 m lap in 80 s has an average speed of 5 m/s. The average velocity is 0 m/s because the displacement is 0 m. OpenStax confirms that a round trip produces zero displacement and zero average velocity.
What Are the Units of Velocity?
The SI unit of velocity is meters per second (m/s). The dimensional formula of velocity is length divided by time, written [L T⁻¹]. The table below converts common velocity units to m/s.
| Unit | Symbol | Equivalent in m/s |
|---|---|---|
| Meter per second | m/s | 1 |
| Kilometer per hour | km/h | 0.2778 |
| Mile per hour | mph | 0.4470 |
| Foot per second | ft/s | 0.3048 |
| Knot | kn | 0.5144 |
Multiply km/h by 0.2778 to get m/s. Multiply m/s by 3.6 to get km/h. A speed of 90 km/h equals 25 m/s.
How Does a Graph Show Velocity?
The slope of a position-time graph shows velocity. The list below summarizes 4 graph relationships.
- Slope of a position-time graph represents velocity.
- Slope of a velocity-time graph represents acceleration.
- Area under a velocity-time graph represents displacement.
- Horizontal line on a position-time graph represents zero velocity.
How Is the Velocity Equation Related to Calculus?
Velocity is the derivative of position with respect to time, written v = dx / dt. Position is the integral of velocity, written x = ∫v dt. Velocity from acceleration follows v(t) = ∫a(t) dt + C, according to the OpenStax key equations.
A position function x(t) = 3t² + 2t, with x in meters and t in seconds, gives v(t) = 6t + 2. At t = 4 s, the velocity is 26 m/s.
How Do You Find Velocity From Other Quantities?
Velocity follows from energy, momentum, flow rate, acceleration, or angular velocity with 5 additional equations. The table below lists each equation with a worked result.
| Known Quantities | Equation | Example Values | Result |
|---|---|---|---|
| Kinetic energy (KE) and mass (m) | v = √(2KE / m) | KE = 200 J, m = 4 kg | 10 m/s |
| Momentum (p) and mass (m) | v = p / m | p = 30 kg·m/s, m = 6 kg | 5 m/s |
| Flow rate (Q) and area (A) | v = Q / A | Q = 0.02 m³/s, A = 0.005 m² | 4 m/s |
| Acceleration (a) and time (t) from rest | v = at | a = 4 m/s², t = 5 s | 20 m/s |
| Angular velocity (ω) and radius (r) | v = ωr | ω = 3 rad/s, r = 2 m | 6 m/s |
What Mistakes Occur When Using the Velocity Equation?
Four mistakes cause most errors in velocity calculations.
- Confusing distance with displacement: distance measures path length, and displacement measures net change in position.
- Mixing units: kilometers with seconds, or miles with minutes, produce incorrect results.
- Ignoring signs: a negative displacement produces a negative velocity.
- Applying constant acceleration equations to variable acceleration: these equations require a constant value of a.
Where Is the Velocity Equation Used?
The velocity equation applies to 5 main fields.
- Transportation: calculating vehicle velocity from position changes over time
- Aerospace: describing orbital motion, where the International Space Station travels at about 7.9 km/s, according to ScienceABC
- Sports science: measuring sprint velocity from split times
- Meteorology: reporting wind velocity in m/s with a direction
- Engineering: determining fluid velocity in pipes with v = Q / A
What Do People Also Ask About the Equation for Velocity?
What is the formula for velocity?
The formula for velocity is v = Δx / Δt, which divides displacement by elapsed time.
What is the equation for final velocity?
The equation for final velocity is v = v₀ + at, if acceleration is constant.
What is the equation for instantaneous velocity?
The equation for instantaneous velocity is v = dx / dt, the derivative of position with respect to time.
What is the velocity equation without time?
The velocity equation without time is v² = v₀² + 2aΔx, if acceleration is constant.
How do you find velocity from distance and time?
Divide distance by time to get average speed, and divide displacement by time to get average velocity.
What is the equation for velocity from kinetic energy?
Velocity from kinetic energy is v = √(2KE / m).
What is the SI unit of velocity?
The SI unit of velocity is meters per second (m/s).
Can velocity be negative?
Velocity is negative when motion is opposite to the chosen positive direction.
What is the escape velocity of Earth?
Earth's escape velocity is about 11.2 km/s at the surface.
What is the velocity of a free-falling object after 1 second?
A free-falling object released from rest reaches 9.8 m/s downward after 1 second, if air resistance is ignored.
What Is the Core Equation for Velocity?
The core equation for velocity is v = Δx / Δt. Velocity is a vector quantity measured in m/s. Speed uses distance, and velocity uses displacement. Four constant acceleration equations extend the core equation to final velocity problems. Calculus defines instantaneous velocity as v = dx / dt. The remaining formulas cover circular, wave, relative, orbital, escape, and terminal velocity.
SEO title: Physics C Equation Sheet: Contents and 57 Study Equations
The University of the West Indies
My passion as a teacher and leader in education is to contribute to students’ enhanced learning by creation of a positive teaching/learning environment, where they are motivated to develop strong critical thinking, analytical and problem-solving skills.
As acting Vice-Principal and head of Physics, I have amassed five years of experience in leading meetings, implementing strategies to maintain discipline and school ethos, liaise with stakeholders in planning for future development of the institution; successfully negotiated the implementation of various projects for student development as well as supervising effective execution of such. For the past 19 years, I have incorporated discovery learning within the classroom through practical activities, problem solving and cooperative learning and have realized that students retention in Physics and Mathematics have increased significantly.
I am always cognizant that students have different learning styles, and as such teaching strategies during planning and delivery must take this crucial factor into consideration and be tailored to meet such differences.