Velocity Formula: Definition, Formulas and Solved Examples

The velocity of an object refers to the rate of change of the displacement of an object with time. It is a measurement of how faster or slower an object is moving. 

• Velocity is a vector quantity i.e. it has both magnitude and direction.
• The SI unit of velocity is meter-per-second (m/s).
• The other unit of velocity is kilometre-per-hour (km/h), miles-per-hour (mph), etc.
• We can also say that the speed of a body along a particular direction is called velocity.
• Speed is a scalar quantity and it only has magnitude.
• Therefore, velocity is the speed of the body moving in a particular direction.

Key Terms: Speed, Acceleration, Motion, Instantaneous velocity, Initial and final velocity, Equations of motion, Angular velocity, Escape velocity, Vector quantity.

Also Read: Relative Velocity

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What is Velocity?

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The velocity of an object is its rate of change in position as observed from a particular frame of reference with respect to time. It is a branch of classical mechanics that describes the motion of the body.

• Since velocity have both magnitude and direction, therefore it is a vector quantity.
• The scalar value or magnitude of velocity is known as speed.
• The velocity of the body changes if either its magnitude or direction or both will change.
• The rate of change in velocity is called acceleration.
• If there is a change in the magnitude of velocity but the direction remains constant then the overall velocity vector of the body changes that leads to the production of acceleration. For example, a car moving in a straight path.
• If there is a change in the direction of the body but the magnitude of velocity remains constant, then also overall velocity vector of the body changes, and acceleration is produced. For example, a car moves in a circular path with constant speed.
• For a body to have a constant velocity, it has to move in a straight path with constant speed.
• Velocity is a derived physical quantity and its SI unit is meters-per-second (ms^-1 or m/s).
• The dimensional formula of velocity is [M⁰LT^-1]

 Speed and Velocity

Also Read: Speed & Velocity

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Velocity Formula

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Consider a body is moving in a straight line with velocity v. Let it is at position x₀ at time t₀, after time t the reaches position x, then the displacement of the body is given by

Δx = x – x₀

And, the time taken by the body to change its position is given by

Δt = t – t₀

Therefore, the velocity of the body (also called average velocity) with which the body moves from position x₀ to x is given by the rate of change of its position and is represented as:

\(v=\frac{x-x_0}{t-t_0}=\frac {\Delta x}{\Delta t}\)

Instantaneous velocity

Instantaneous velocity gives the velocity of a particle at a particular instant of time. The instantaneous velocity of a body at a given instant of time is defined as the limit of average velocity as the time interval Δt becomes infinitesimally small. It is given as

\(v_{ins}= \lim_ {\Delta t\to 0} \frac {\Delta x}{\Delta t}=\frac{dx}{dt}\)

Where dx/dt is called the derivative of x with respect to t.

Initial and Final Velocity

The initial velocity of a body refers to the velocity at time t = 0 with respect to a frame of reference. It may be zero if the body is at rest or have some finite value at t = 0. Whereas the final velocity of a body refers to the velocity at time t = t with respect to a frame of reference. It may be zero if the body after travelling some distance comes to rest.

The initial and final velocity of a body can be calculated using equations of motion:

1. S = ut + 1/2 at²
2. v² = u² + 2as
3. v = u + at

Where

• u = initial velocity of the body
• v = final velocity of the body
• a = constant acceleration
• t = time taken
• S = displacement of the body

Also Read: Velocity Selector

Angular Velocity

Angular velocity is defined as the velocity of a body moving in a circular path. It is the rate of rotation of the body moving in a circular path about a fixed point.

• SI unit of angular velocity is radian-per-second (rad/s or rad s^-1).
• Mathematically, it is given by the change in position about a fixed point in a circular path or angular displacement divided by the total time in which the change in angular position takes place. i.e.

\(\omega=\frac {d\theta}{dt}\)

Where

• ω = angular velocity
• dθ = change in angle of rotation or angular displacement
• dt = time in which angular displacement takes place

Also Read:  Value of c: Speed & Velocity of light, Importance, Consistency

Escape Velocity

Escape Velocity refers to the minimum velocity required by an object to escape from the gravitational influence of a massive body. It is given by

\(v_e= \sqrt {\frac {2GM}{R}}=\sqrt {2gR}\)

Where 

• v_e = Escape Velocity 
• G = Universal Gravitational Constant 
• M = Mass of the massive body
• r = Distance of the object from the centre of mass of the massive body

The escape velocity of Earth is 11.2 x 10³ m/s or 11.2 km/s.

Also Read: Derivation of Drift Velocity

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Difference Between Speed and Velocity

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A detailed comparison between speed and velocity is given below:

Also Read: Drift  Velocity

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Solved Examples

Ques. A car travelling in a straight road 60 km in 3 hours. What is the Velocity of the car?

Ans. Given,

• displacement of the car, s = 60 km = 60000 m
• time taken, t = 3 hours = 3 x 60 x 60 = 10800 seconds 

We have,

Velocity = s/t 

= 50000 / 7200 

= 5.56 m/s

Ques. Calculate the final velocity of a car if it starts from rest and travels for 20 sec with a constant acceleration of 4 m/s².

Ans. Given

• The initial velocity of the car, u = 0 m/s 
• Acceleration, a = 4 m/s² 
• time, t = 20 sec

From the equation of motion, we have

Final Velocity (v) = u + at

= 0 + 4 x 20

= 80 m/s

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Things to Remember

• The velocity of an object refers to the rate of change of the object’s position to a frame of reference.
• Velocity is a vector quantity. 
• The scaler counterpart of Velocity is called Speed.
• The SI unit of Velocity is metres per second (m/s).
• Angular Velocity refers to how fast an object is rotating to another point.
• Average Velocity refers to the constant velocity that would provide the same resultant displacement as a variable velocity over a certain time period.
• Escape Velocity refers to the minimum velocity required by an object to escape from the gravitational influence of a massive body.
• Initial Velocity refers to the velocity at time t=0 and is denoted by ‘u’. It is also the velocity at which an object starts its motion.