Average Velocity Formula

Average velocity is the expression of total displacements possessed by a body in time ‘t’. So, if a body travels various displacements viz: s₁, s₂, s₃,......,n in a given time ‘t’, then finding average velocity becomes easy by using the following average velocity equation:

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In SI systems, the unit for the average velocity is the same as that of the velocity, i.e., m/s. In this article, we will derive the formula to find average velocity and also the average speed equation.

Average Speed Formula Derivation

Let’s suppose that a particle travels varying distances viz: s₁, s₂, s₃, s₄,...., sn in their respective velocities viz: v₁, v₂, v_3,....,v_n in the same direction, so we know that the time is equivalent to the division of distance by the speed, for this scenario, we have the following formula:

Total time taken ‘t’  = \[\frac{s_1}{v_1}\] + \[\frac{s_2}{v_2}\] + \[\frac{s_3}{v_3}\] + ….. + \[\frac{s_n}{v_n}\]

We know that the speed = distance/time, so here the formula for the average speed goes:

Average speed v_av = \[\frac{s_1 + s_2 + …..}{(\frac{s_1}{v_1} + \frac{s_2}{v_2} + ……)}\] 

Now, if we say the varying distances are the same and the body travels with two varying speeds v₁ and v₂ over two equal distances, i.e., s₁ = s₂, the formula becomes the following:

= \[\frac{2s}{(\frac{s}{v_1} + \frac{s}{v_2} + ……)}\]

= \[\frac{2v_1 v_2}{(\frac{1}{v_1} + \frac{1}{v_2})}\]

vav = \[\frac{2v_1 v_2}{(v_1 + v_2)}\]

This expression clearly describes us with an idea that average speed is the equivalent of the harmonic mean of individual speeds.

Average Speed Formula in Physics Derivation

If we say that a man travels to his favorite hilly area with varying speeds viz: v₁, v₂, v₃,...., and so on during time intervals viz: t₁, t₂, t₃,..., and so on, respectively. 

So, the distance travelled will be: v₁t₁ + v₂t₂ + v₃t₃

Time taken in total is t₁ + t₂ + t₃ +.....

So, the average speed by which he reached his journey is given as:

vav = \[\frac{v_1t_1 + v_2t_2 + v_3t_3}{t_1 + t_2 + t_3+.....}\] 

If we suppose that the time taken in each interval is the same, i.e., t₁ = t₂ = t₃ =,.....= t_n, then we get the new equation in the following way:

vav = \[\frac{v_1 + v_2 + v_3...}{n}\] 

We notice that the average speed is the arithmetic mean of the individual speeds.

Unit of Average Speed

Here, if you determine the unit of the average speed, it comes out in the following manner:

= \[\frac{2(m/s)(m/s)}{(m/s + m/s)}\]

Canceling the common terms, we get the unit as m/s again.

Dimensional Formula of Average Speed

The dimensional formula for the average speed is as follows:

= \[\frac{2(LT^{-1})(LT^{-1})}{(LT^{-1}+LT^{-1})}\]

The dimensional formula for the average speed comes out as LT\[^{-1}\].

Finding Average Speed Solved

Let’s suppose that you and your friend decide to go on a bike ride to a different state, i.e., Kerala. 

Your bike is a duke and your friend’s bike is BMW. Your bike can travel at a speed of 300 kmph, while your friend’s at the speed of 500 kmph, now, the average speed formula Physics says that if the two speeds are given then we can determine the average speed, and that is given by:

v_av = \[\frac{2v_1v_2}{v_1+v_2}\] 

= \[\frac{2 \ast 300 \ast 500}{300 + 500}\] = 375 kmph

So, the average speed comes out as 375 kmph, so this was another formula or you can say the short trick to determine the average speed. 

Now, we have something very interesting besides speed (a scalar quantity) and that is velocity, so do you know what is velocity?

Average Velocity Equation

A velocity is something that is the division of the displacement of an object by time. In simple terms, it is the displacement possessed by an object in a unit of time. we can determine the average velocity of the object’s journey.

If we know the initial velocity and the final velocity of an object, we can determine the average velocity as:

v_av = \[\frac {(u + v)}{2}\]

Here,

u = initial velocity of the object and v is its final velocity. Both of these are measured in m/s and their mean is the average velocity, i.e., vav. The average velocity is also measured in m/s and its dimensional formula is LT\[^{-1}\] .

Another equation for the average velocity is:

         =  (Final position  - initial position)/(end time - starting time)

If we have to calculate the velocity at an instant, then the formula turns to the instantaneous velocity formula. 

Finding Average Velocity

Example:  A boy walks with an initial velocity of 40 m/s and he reaches his destination at 90 m/s. Calculate its average velocity. 

We are given the following data:

Initial Velocity u = 40 m/s

Final velocity v = 80 m/s

The formula to find average velocity is as follows:

vav= \[\frac {(u + v)}{2}\]

So, our answer comes out as:

Average velocity vav = \[\frac {(40 + 90)}{2}\] = 65 m/s.

Conclusion

We have seen various formulas for calculating the average speed and average velocity for different situations. The average speed formula helps in determining the average value of speed if the body is varying continuously for the given time intervals. velocity can be positive or negative depending on the direction we are choosing as positive while speed always remains positive.