Average Velocity

What is Average Velocity - Definition and Formula

The term motion can be described by using some physical quantity terms like speed, velocity, distance, displacement, and acceleration. The proper description of motion was given by Sir Isaac Newton. All these quantities are described with respect to one parameter, that is time. Here, we discuss average velocity and its mathematical representation along with its graphical representation. 

Average velocity definition

The word average simply means, the ratio of the sum of quantities to the total number of quantities. In physics, a different approach is taken. Now before understanding average velocity, let us understand what velocity exactly is and what speed is and how these two are connected.

The speed on an object refers to the change in position of that object with respect to time. Velocity on the other hand is nothing, but the speed defined with respect to the direction in which an object travel.

We will dive in detail about the difference between speed and velocity, for now, let's come back to average velocity.

Now average velocity according to the definition is the ratio of the displacement from point a to point b of an object to the time it had taken to make that displacement from point a to point b. It may be noted that we use the term movement instead of distance to emphasise the direction. 

Algebraically an average velocity is defined as, v = d/t

where d is the displacement and t is the time taken for that displacement.

For a short interval of time, the average velocity can be calculated as follows:

va = [(y0+Δy) −y0)/[Δt]]

where y0 is the position of an object at time t and (y0+Δy) is its position in the same direction after an increase of time by Δt.

When we take the limit as Δt→0, then it becomes dy/dt, the average velocity turns into instantaneous velocity at time t.

When an object undergoes a change in velocities at different time, the average velocity is determined by the total sum of all the velocities at different instances divided by the number of total time. For example, if an object has different velocities v1, v2, v3, vn, at times t = t1, t2, t3, tn, then the average velocity is given by,

va = [v1+v2+v3......+vn]/ [n].

Difference between average speed and average velocity.

Previously we explained what speed is and what is velocity and what is the difference between the two.
Speed is a quantity which is scaler and velocity is a vector quantity.

Now, for example, we talk about a car running at a speed of 80 miles per hour with no reference to its direction then we are talking about its speed. Whereas if we say that a car is running at a speed of 80 miles per hour towards the northeast then we are referring to its velocity. Time will always be a scalar quantity and the direction of displacement will be the direction of velocity.

Let us take another example to get a better understanding of this concept.

Take for example a car is travelling towards the east at a speed of 80 km per hour for 2 hours and at the same speed for one hour but in the direction of west, which is opposite direction to earlier.

Now the total distance travelled is 80 × 2 + 80 × 1 = 180 km and the total time has taken is 3 hours. Therefore, the average speed is 1803 = 80 km per hour. But when you calculate the displacement of the car, the net displacement is 80 ×2 – 80 × 1 = 80 km towards east. Therefore, the average velocity is 803 = 20 km per hour towards the east.

Average Velocity Formula

When we take out the average value of a given number of velocities it is known as average velocity. Average velocity is the displacement of an object over time. To find the average speed of an object we divide the distance travelled by the time elapsed. We know that velocity is a vector quantity and average velocity can be found by dividing displacement by time.
The units used for velocity can be derived from its definition i.e. meters/second (standard SI unit) or any distance unit upon any time unit.

If the velocity of a body is varying continuously for a given period of time it can be very helpful to determine the average velocity of that object to get an idea of its total journey.

Average Velocity Formula mathematical description

It is given by Vav. The formula of average velocity involves total displacement and total time taken for that displacement.

For any positions, xi and xf along with their corresponding time intervals ti and tf are specified, we use the formula that is given below
\[{v_{av}} = \frac{{{x_f}--{x_i}}}{{{t_f}--{t_i}}}\]
Where xi = Initial position,
xf = Final position,
tf –ti = time interval

Finding Average velocity

To calculate average velocity first we need to find out the net displacement of a given object during the course of its entire motion. The initial direction of the motion of the object indicated in the question is taken as its direction for the entire course until specified.

 The following diagram will make the concept of average velocity clearer.

Suppose an object is traveling at a distance d1 in a given time t1, d2 in a time t2, and d3 in a time t3, as shown above. It should be well-known that the distances travelled are not in similar directions. d1 + d2 + d3 is the total distance but it is not the net displacement. The net displacement is the projections of d2 and d3, in the direction of d1, are d2 + d3 and hence the net displacement is d1 + d+ d3. Therefore, in the above- given case the average velocity can be found by,

Vav = [d1+d2+d3]/ [t1+t2+t3]

and in general,

Vav= [d1+d2+.... +dn]/ [t1+t2+.... +tn].

One thing to be noted here is that if the velocity formed in the diagram is taking the shape of an obtuse angle then the said velocity will come out in negative values.

The magnitude of average velocity
Vector quantities always have a direction and magnitude and since we have defined velocity as a vector quantity, therefore, it has both magnitude and direction. If in any case the direction of an object is ignored, the datum of the average velocity is taken the magnitude of the average velocity. One thing to remember here is that, while calculating an average velocity for a given object where a data of velocities has been given for different time intervals, you need to ignore the direction not at computation but only at the final stage.

Now recall the formula we derived for taking out average velocity in the last section.

Average Velocity

For an object that has n number of velocities in n number of consecutive time intervals, the magnitude of the average velocity can be given by,
Magnitude of Vav = [d1+d2+.... +dn] /[t1+t2+.... +tn]

Average angular velocity

Till now we have been discussing so far about the average velocities in case of linear motions, i.e., objects moving away from or towards a given point in the straight line. But learning about circular motion is equally important. In a circular motion, an object moves around a point in a circular path. The best example of this is the motion of rotating wheels.

In case of motion which is circular, a term called angular velocity can be defined. An angular velocity can be measured in terms of the angle covered by the object in circular motion per unit time. We normally denote an angular velocity by the Greek letter ω. The direction of angular velocity is limited to a clockwise or anti-clockwise direction. The datum will be referred to as angular speed in case of the absence of the direction.

Therefore, the definition of average angular velocity is:

ω = θ/t,
where θ is the angle rotated in the time t

When we talk about angular velocity, there are only two possible outcomes, the computation of average velocity is simpler. It is either of the two- positive or negative. As a convention, counter clockwise direction is considered as positive and the clockwise direction as negative. The basic unit of angular velocity is radians per unit time, mostly, radians are per second. It may be kept in mind that one revolution means covering an angle of 2π radians.

In this case also, when different angular velocities occur at different times, the average angular velocity is found by this given formula.

ωav = [θ1+θ2+.... +θn] / [t1+t2+.... +tn] (if time intervals are consecutive.