Measuring the Rate of Change of Motion

According to Newton’s first law of motion: A body tends to remain at rest or in motion until or unless an external force is applied to it.

Therefore, on giving a push to a body, the body changes its position.

So,the change in position of a body over time is called the motion. 

There are various units to measure the rate of change of motion of a body namely:

speed, average speed, velocity, average velocity, instantaneous velocity, etc.

Here, the basic consideration for the measurement of rate of change of motion is time.

Let’s discuss these units in detail.


Speed


What do you Understand by the Term, ‘Speed’?

Speed of any body is defined as the rate of change of position of that body in any direction.

It is also defined as the distance covered by the body in a unit time in any direction. 

The formula for the same is given by:

                                     Speed = Distance Covered / time taken  = D/T

It’s unit is m/s in the mks system or SI and cm/s in the cgs system.


Types of Speed:

  1. Uniform speed: An object that covers equal distances in equal intervals of time.

  2. Variable speed:  An object that covers unequal distances in unequal intervals of time.             

It has a magnitude because the direction of the body is unknown.

Therefore, speed is a scalar quantity.


Average Speed

When an object is moving with a variable speed, then the average speed of that object is that constant speed with which the object covers the same distance in a given time, as it does while moving with variable speed in a given time.


Average Speed Formula Physics

Average speed formula in physics is given by:               

Total distance covered/ total time taken = 2 v1 v2/ (v1 + v2)


Its unit is also m/s in the mks unit and cm/s in the cgs system.


Define Average Velocity

Velocity of an object is equivalent to the rate of change of displacement.

Where displacement is a vector quantity and is defined as the distance between the two positions of the object in a particular direction during a given time.

Velocity is a vector quantity as it has both magnitude (speed) and direction.

The value of velocity and displacement can be expressed as zero, negative or positive.

The average velocity is defined as the total displacement of the body divided by the total time taken.


Average Velocity  Physics

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Fig. Displacement-time graph

If you look at the above graph, there are two points A and B.

Here, the point A corresponds to time t1 and displacement x1 while point B to time t2 and displacement x2.

Displacement of an object at time interval (tᵣ - tᵢ) =  xᵣ - xᵢ.

The formula for the average velocity is given by:           

  Vav =  Total displacement /Total time = Δx /Δt = xᵣ - xᵢ/ tᵣ - tᵢ..(1)


 Here, 

              xᵢ = initial distance

              xᵣ =final distance

               tᵢ  = initial time

               tᵣ =  final time

If there are diverse distances let’s say d1, d2, d3….dn in different time intervals t1,t2,t3,…..tn     

Then, formula is given by,

Vav  =  (d1+d2+d3+....+dn)/ (t1+t2+t3+....+tn)


The slope of a straight line AB is given by:                  

Vav =   xᵣ - xᵢ/ tᵣ - tᵢ = tanΘ…(2)


The eq(2) states that the magnitude of average velocity of the object between two points is equal to the slope of the straight line AB joining these two points on the graph.

Average velocity formula physics

  If   u = initial velocity

       v  = final velocity

Then the formula for the mean velocity is given by,                                

Vmean  = (u + v)/2


The average velocity equals the sum of the initial velocity and the final velocity divided by 2.


Instantaneous Velocity Physics

The word instant means at that very moment. 

So the instant velocity of an object in motion is calculated at a specific point(x,t).

This velocity is the limit of average velocity as the elapsed time approaches zero.

Since average velocity between two points A and B is

                                  Vav= xᵣ - xᵢ/ tᵣ - tᵢ

If time interval is small i.e., t2 -t1 = Δt and x2 - x1 =  Δx

 If Δt -> 0, the average velocity becomes instantaneous velocity.

Then,                       

  Vins = lim    Δx/Δt   = dx/ dt

             Δt-> 0


Hence, instantaneous velocity of the object is the first derivative of displacement with respect to time. 


Do you know?

When a body is traveling through a uniform motion along a linear path in a given direction, the magnitude of the displacement is equivalent to the actual distance traveled by the body in a given time.

FAQ (Frequently Asked Questions)

Q1: How do you Find Average Speed?

Ans: Let’s take an example:

Suppose a truck travels a distance from A to B at a speed of 60 kmph and returns to A at a speed of 120 kmph. Average speed of the whole journey is:

Let AB = s,  time taken from A to B = t1 = s/60

Then, BA = s  and t2 = s/120

Total time taken = t1+ t2 = s(1/60 + 1/120) = s/40 h

Total distance = s + s =2s

So,  average speed will be:

 Vav = Total distance traveled/ total time taken = 2s/ s/40 = 80 kmph.

Q2: What is an Example of Average Velocity?

Ans:  A woman takes 60 minutes to drive 30 miles east and then 30 miles west to end up at the same place.

She walks with an average speed of 60 miles divided by 60 minutes equal to 1 mile per minute or 60 mph.

Q3: Derive the Average Velocity Formula.

Ans: Let’s consider the equation of 2nd law of motion : 

s = ut + ½ at²

Since v = s/t 

v = (ut + ½ at²)/t

   = u + at/2

   = (2u + at)/2

   = ((u + (u + at)/2 …..(1)

From first law of equation of motion: v = u + at putting in eq (1)

Average velocity = u + v/ 2 = (vᵣ + vᵢ)/2