Velocity 

The velocity of any object refers to the speed with which it is moving in a given direction. It tells you how fast something is moving and in which direction. Unlike speed, velocity includes both the object's speed and the direction it travels.

1.0Basics of Velocity 

Velocity (represented by “v”) is a vector quantity, meaning it has both magnitude and direction. It describes how fast an object is changing its position. Mathematically, velocity is the rate of change of a position, called displacement, with respect to time in a given direction. Velocity in mathematical terms can be represented as: 

$v=\frac{dr}{dt}=\frac{Displacement}{time}$

Here:

• v represents the velocity vector,
• r represents the position vector,
• t is the time.

The SI unit of Velocity, just like speed, is m/s or ms^-1.

2.0Types of Velocity 

Uniform Velocity

Uniform Velocity is the constant velocity over a certain distance meaning an object in motion covers an equal distance in an equal interval of time without changing its direction to cover that distance. For Example, a car on a journey of 5km covers every 500 meters with a velocity of 25m/s.

Non-Uniform Velocity

It refers to the velocity when an object covers a different/same distance at the same/different speed and/or changes its direction throughout the journey. For example, a car accelerates or decelerates along a curved path.

Initial Velocity

Initial velocity refers to the velocity with which an object or vehicle starts its journey from rest to motion or motion to motion. It is commonly denoted with the letter u. When not given the initial velocity is always taken 0.

Final Velocity

The final velocity is the velocity at the end of the journey or the end of the time interval of an object in motion. Usually, it is denoted with the letter v. 

Linear Velocity

Linear velocity is referred to as the speed of any object that is moving in a straight line. It is the speed at which the position of an object in a straight line changes with time. The formula for linear velocity is: 

$v=\frac{s}{t}$

Where v = velocity, s = displacement, t = time. 

Linear velocity is simply the speed along a straight path when there is no change in its direction. 

Angular Velocity 

Angular velocity is how fast an object is going about a fixed axis, and as such, the word is commonly used when dealing with circular or rotational motions. It measures how speedily an angle changes over time. The Formula: 

$\omega =\frac{\Delta \theta }{\Delta t}$

Here:

• ω is the angular velocity,
• Δθ is the change in angle (in radians),
• Δt is the time interval.

Instantaneous Velocity

Instantaneous Velocity is the velocity at the exact moment in the whole time interval. Imagine a speedometer of a car or bike shows a speed of 30km/h, it is the speed of car or bike at that particular moment, this velocity at a particular moment is the Instantaneous Velocity. Instantaneous velocity is given by: 

$\omega =\frac{\Delta \theta }{\Delta t}$

Here, 

ds = change in position

dt = change in time 

v = Instantaneous velocity.  

Average Velocity 

Average velocity can be defined as the total displacement divided by the total time taken for a given motion. It is a vector quantity; which means it carries with itself both magnitude and direction. The formula for average velocity is: 

$v_{avg}=\frac{TotalDisplacement}{totaltimetaken}$

3.0Relative Velocity 

The term relative velocity is defined as the velocity of one moving object with respect to another. If the two objects move in the same direction, their relative velocity is just the difference of their velocities. However, if they are moving in opposite directions, their relative velocity will be a sum of their velocities. 

For example, if one car is moving at 50 km/h and another at 30 km/h in the same direction, their relative velocity is 50−30 = 20 km/h.

4.0Key Differences

Difference Between Speed and Velocity 

Difference Between Velocity and Acceleration 

5.0Distance-Time Velocity-Time Graphs

Distance-Time Graph 

The slope represents the speed or velocity of an object. A straight line indicates uniform motion, shown below and a parallel line to the x-axis shows the object to be at rest(like 1 to 2 hours), while a curved line indicates non-uniform motion. 

Velocity-Time Graph 

In this graph, the slope represents acceleration, and the area under the curve is the displacement of the object. If the velocity is constant, then the acceleration will be zero, it is a horizontal line. If the velocity increases or decreases, then the graph will have a slope that indicates acceleration.

6.0Velocity in Different Contexts 

Relation between Frequency and Velocity

In wave motion, frequency is the number of complete cycles or oscillations a wave makes per second. It is measured in Hertz (Hz). The velocity v of a wave is related to its frequency f and wavelength \lambda in the following way: 

$v=f.\lambda$

Relation between Pressure and Velocity 

In fluid dynamics, Bernoulli’s Principle gives a relationship between pressure and velocity. According to this principle, for an incompressible, non-viscous fluid, the pressure and velocity are inversely related as: 

$P+\frac{1}{2}\rho v^2+\rho gh=constant$

Here:

• P is the pressure,
• ρ is the fluid density,
• v is the velocity,
• g is the acceleration due to gravity,
• h is the height.

7.0Velocity Questions 

Question 1: A car travels from point A to point B, covering a distance of 120 km in 2 hours. Then, it returns from point B to point A in 3 hours. What is the average velocity of the car for the entire trip?

Solution: Total displacement = 0km as the car returns to the starting point. 

Total time = 2 + 3 = 5 hours. 

Using the formula for average velocity: 

$v_{avg}=\frac{TotalDisplacement}{totaltimetaken}$

$v_{avg}=\frac{0}{5}$

$v_{avg}=0km/h.$

Question 2: A wheel rotates through an angle of 2π radians in 4 seconds. What is its angular velocity?

Solution: change in angle $\Delta \theta =2\pi$, time taken t = 4s 

$\omega =\frac{\Delta \theta }{\Delta t}$

$\omega =\frac{2\pi }{4}$

$\omega =\frac{\pi }{2}rad/s$

Question 3: Car A is travelling at 30 m/s to the east, and Car B is travelling at 20 m/s to the east but in the opposite direction (to the west). How far apart are the two cars after 10 seconds? 

Solution: Let the velocity of car A = $v_A$ = 30m/s

Let velocity of car B = $v_B$ = 20m/s

Since the cars are moving in opposite directions the relative velocity of cars will sum up = 50 m/s

Using the formula: 

$v=\frac{s}{t}$

Distance covered in 10s 

$50=\frac{s}{10}$

s=500m

8.0Also Read