How to find instantaneous velocity from average velocity: Detailed Insights


First and foremost, before knowing how to find instantaneous velocity from average velocity, we have to know about average and instantaneous velocity.

The instantaneous velocity is considered the measure of a limit of Vavg as the time tends towards a specific value. Average and instantaneous velocity differ as their name suggests; one is used to calculate total velocity, and the other is used to measure instant velocity.

We’ll know the fundamental differences of the above two velocities before learning how to find instantaneous velocity from the average velocity.

Average velocity: Points to remember

Before going to the central concept, let us first know about the average velocity.

  • Average velocity is one of the primary concepts used in physics to measure how much distance a particle covers in its motion during a specific period.
  • It is one of the main scalar physical quantities.
  • It has a dimension equal to length/time since it is measured using the area covered in a certain period.

Now let’s have a look at instantaneous velocity

Instantaneous velocity: Points to remember

After knowing about average velocity, now it is mandatory to know about instantaneous velocity.

  • Instantaneous velocity is the fundamental concept of physics that is used to find out the velocity of a particle, which is in motion at any particular point of time, using the position and time of that particle.
  • It is a primary vector quantity.
  • Even it has a dimension similar to average velocity [ Distance/time = L/T].

Now let’s know how to find Vinst from Vavg, which is the post’s focus.

How to find instantaneous velocity from average velocity

Before measuring the instantaneous velocity of an object, it is necessary to find out its average velocity.

  • In a distance-time graph, first, we have to measure average velocity by taking the two increments, i.e., one larger and one smaller increment of distance and time to a point for which we will measure Vinst.
  • After this, we have to take the average of those smaller and larger points.
  • For this value, you have to consider the Vavg formula and apply its limit at t equals any value of your choice. You will be able to find Vinst.

Now let us know in detail how to find instantaneous velocity from average velocity by using the formula.

From formula

We will know how to find Vinst from Vavg step by step as given below.

Step 1

Take a displacement equation of the form S.

Step 2

Consider the given two-time intervals for which you have to find the average velocity.

Step 3

Try to calculate the average velocity for those values with the help of the formula given below,

It can also be written as below,

Step 4

If you want to calculate instantaneous velocity, you have to apply a limit function to the average velocity formula.

how to find instantaneous velocity from average velocity

Now let us know in detail how to find instantaneous velocity from average velocity by using a graph.

From graph

The most straightforward method to find Vinst from Vavg is from the representation of the graph.

Step 1

Use both the axes of the graph to represent position and time respectively on the x and y-axis.

how to find instantaneous velocity from average velocity

Step 2

Consider any displacement equation and insert specific values for t and mark it as follows on the graph.

                                                                  [ t, S (X, Y)]

Now consider any two points as (x, y) and (x1, y1) and name them as C and D.

how to find instantaneous velocity from average velocity

Step 3

In step 3, you have to connect those two points through a line and measure the average velocity; one can use the below formula to calculate average velocity.

                 

how to find instantaneous velocity from average velocity

where K refers to the slope between C and D.

Step 4

The line drawn to the slope taken is considered to be a tangent line.

Repeat the same process as mentioned in step 3, and continue choosing the points closer to each other.

After several attempts, it reaches a single point on the tangent line. Repeat to find slope several times, moving D nearer to C. 

how to find instantaneous velocity from average velocity

If we take the limit to that single point, we can find out the value of the slope at that particular point.

Step 5

Now it is time to solve some problems based on average and instantaneous velocity to understand them better.

Problems to find instantaneous velocity from average velocity

The problems that are solved below are some of the basic calculations to find Vinst from Vavg .

Problem 1

Find the average velocity equation for the given displacement function S = -6t2 + 11t + 9 and even calculate the instantaneous velocity at t=7s?

Solution:

The given displacement equation is as follows

S = -6t2 + 11t + 9

ds/dt = -(2) 6t (2-1) + (1)11t1 – 1 + (0)9t0

 ds/dt  = -12t1 + 11t0

 ds/dt = -12t + 11

This will be the equation to find the average velocity. Now, if we substitute the limit value t tends to 7s, we can calculate instantaneous velocity.

 ds/dt = -12(7) + 11

 ds/dt = -84 + 11

 ds/dt  = -73 meters/second

Therefore, the instantaneous velocity is -73m/s.

Problem 2

The bus covers a distance of 48Km in 45 minutes, and the equation for the motion of the bus is given by the function S = 6t2 + 8t + 3. Find its average velocity and then compute its Instantaneous Velocity at time t = 8s?

how to find instantaneous velocity from average velocity
Image Credit:
Pixabay free images

Solution:

First let us find average velocity

ds/dt = 48/45

Vavg = ds/dt = 1.066 m/s

The function is S = 6t+ 8t + 3.

Differentiate the given equation with respect to t, then we get

ds/dt = 12t + 8

For time t = 8s, the Instantaneous Velocity is articulated as,

V(t) = 12t + 8

V(5) = 12(8) + 8

V(5)= 104 m/s.

Thus for the known function, Instantaneous Velocity is 104 m/s.

Problem 3

Find the average velocity at a given time interval of a kid when he moves 8 m in 5 s and 20 min 8 s along the horizontal axis?

how to find instantaneous velocity from average velocity
Image Credit:
Pixabay free images

Solution: Initial distance traveled by the kid, xi = 8 m,

Final distance traveled by the kid, xf = 20 m

Initial time interval ti = 5 s

Final time interval to = 8 s

Average velocity V = xi − xf / ti − tf = 20−8 / 8−5 = 12 / 3 = 4 m/s

Frequently asked questions | FAQs

What is the formula to measure instantaneous velocity?

The formula used to measure instantaneous velocity can be taken as follows,

As we know, both average and instantaneous velocity can be calculated in the same motion; then, we can find instantaneous velocity by using the same formula as average velocity but with the addition of a limit value.

Which graph is used to find instantaneous velocity from average velocity?

The graph that helps know the velocities of an object is the Position-time graph.

In a position-time graph, we can measure certain different concepts of physics. It is the primary graphs that help to measure instantaneous velocity, average velocity, speed, etc.

Are average speed and average velocity the same?

Both average speed and average velocity are entirely different quantities.

Speed, in general, refers to a scalar quantity that signifies only the magnitude of the object. In contrast, velocity refers to vector quantity that indicates both magnitude and direction of a body in motion. The word average refers to total calculation, whereas instantaneous refers to a particular moment.


Raghavi Acharya

I am Raghavi Acharya, I have completed my post-graduation in physics with a specialization in the field of condensed matter physics. Having a very good understanding in Latex, gnu-plot and octave. I have always considered Physics to be a captivating area of study and I enjoy exploring the various fields of this subject. In my free time, I engage myself in digital art. My articles are aimed towards delivering the concepts of physics in a very simplified manner to the readers. Let’s connect through - LinkedIn: https://www.linkedin.com/in/raghavi-c-s-260a801b1 EMAIL ID: raghavics6@gmail.com

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