Instantaneous Velocity and Speed: Detailed Class 11 Notes

Instantaneous velocity is the velocity of an object in a single instant in time. 

Note that instantaneous velocity is a vector quantity. Because the displacement of an object from one position to another considers the direction.

Average velocity is just the foundation to learn instantaneous velocity.

First, let’s see why average velocity isn’t enough. 

Average velocity cannot specify the rate of motion at a specific moment. It only looks at the change in position (displacement) and time. 

Average velocity = (Final position - initial position)/ (Final time - initial time)
Mathematically,
v̄ = Δx / Δt

v̄ = average velocity (the bar notation indicates average)

Δx = displacement (change in position) = x₂ - x₁

Δt = time interval (change in time) = t₂ - t₁

With this average velocity formula, we cannot know the variations in the rate of motion anywhere between the final and initial positions and times. 

Now, instantaneous velocity allows us to know an object’s speed at the particular instant of time. That’s how instantaneous and average velocity differs. 

Mathematically speaking, instantaneous velocity emerges from average velocity. Consider average velocity to be the starting point. 

The only problem is at any single instant of time, we have Δt to be zero. 

This would make velocity = ∆x/0 = undefined (division by zero)

The solution then is to shrink the time interval. Since we can't use ∆t = 0, we use the next best thing:

• Calculate average velocity over a very small time interval around that instant
• Make the interval smaller and smaller
• See what value the average velocity approaches

And the way to shrink the time that is not zero, is to introduce limits. This is a mathematical concept from calculus, which describes what happens to a value as a variable approaches a specific point. The next section explains this better. 

NCERT defines instantaneous velocity as the “velocity at an instant is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small. In other words, 

$\stackrel{\to }{v}=\underset{\Delta t\to 0}{lim}\frac{\stackrel{\to }{\Delta x}}{\Delta t}=\frac{\stackrel{\to }{d}}{\mathrm{dt}}\stackrel{\to }{x}$

where the symbol lim ∆t→0 stands for the operation of taking limit ∆t→0 of the quantity on its right. In the language of calculus, the quantity on the right-hand side of the equation $\stackrel{\to }{v}=\underset{\Delta t\to 0}{lim}\frac{\stackrel{\to }{\Delta x}}{\Delta t}$ is the differential coefficient of x with respect to t and is denoted by  $\frac{d\stackrel{\to }{x}}{dt}$ .”

This formula for instantaneous velocity is essential for IISER Exam. Do refer to the IAT Physics syllabus for further details. 

One more concept that requires some more explanation is how the 'limit of average velocity' is 'instantaneous velocity'. 

Instantaneous Velocity as the Limit of Average Velocity

You calculate average velocity as displacement divided by time interval: Δx/Δt.

But to find velocity at a specific instant, you can't divide by zero. Because that would mean the time interval is zero.

That’s why we need limits from calculus. In calculus, a limit is basically what value a function gets closer and closer to as you plug in numbers that get closer and closer to some specific number.

Now, in the same way and logic, you would examine average velocity over smaller and smaller time intervals around that instant. This is why we observe what value it approaches as Δt becomes extremely small but never zero.

Mathematically: v = lim(Δt→0) Δx/Δt = dx/dt

This makes instantaneous velocity the derivative of position with respect to time.

There are three ways to represent instantaneous velocity. They have different use cases of their own. 

Method 1: The Graphical Method for Visualising Instantaneous Velocity

If you have a position-time (x-t) graph, the instantaneous velocity at any point is equal to the slope of the tangent line to the curve at that exact point. 

The instantaneous velocity at any point equals the slope of the tangent line at that point.

Blue curve: Shows position vs time

Red line: Tangent line at the selected time point

Green lines: Show how secant lines approach the tangent as time intervals shrink

Slope calculation: The steepness of the tangent line gives the exact velocity at that instant

This method is illustrative but often impractical for precise calculations. Let’s see why. 

If you draw a line between two points on the curve, you get the average velocity between those points.  This gives you the slope over an interval, not at a specific instant.

Method 2: The Numerical Method for Calculating Instantaneous Velocity

Let’s say, you have data of positions at different times. You can approximate the instantaneous velocity by calculating Δx/Δt for progressively smaller time intervals Δt, finding the value it converges to.

Method 3: The Calculus Method - Most Precise Way

If the position x is described by a function of time, x(t), this is the best method.

When you find the derivative of the position function x(t), you are creating a new function. That is the velocity function v(t), which gives you the precise instantaneous rate of change for any moment in time. 

A majority of the Motion in a Straight Line Class 11 NCERT Solutions requires conceptual know-how of all these three methods. But the instantaneous velocity formula for class 11 based on the calculus method is most important. 

Here is a quick table to revise. 

Method	How it Works	Limitation
Graphical	Measures the slope of a hand-drawn tangent line on a graph.	Imprecise. Accuracy depends on how well the graph is plotted, and the tangent is drawn. It's an estimation.
Numerical	Calculates Δx/Δt for a very small but finite Δt.	An approximation. It gets very close to the true value but never technically reaches the "instant" where Δt = 0.
Calculus	Uses differentiation to find an exact formula for velocity v(t) directly from the position formula x(t).	Exact and universal. It gives a perfect formula that works for all points in time, not just one.

Before going further, the NCERT definition of instantaneous speed is
“Instantaneous speed or simply speed is the magnitude of velocity. For example, a velocity of + 24.0 m s⁻¹ and a velocity of – 24.0 m s⁻¹ —both have an associated speed of 24.0 m s⁻¹.”

Instantaneous speed is a scalar quantity, and it has no direction. Remember that. 

So if you are wondering if instantaneous speed has a formula, you can calculate it like this. 
For a particle covering distance $\mathrm{\Delta }s$ in time $\mathrm{\Delta }t$ , average speed is $\frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}$ . As $\mathrm{\Delta }t\to 0$ , instantaneous speed is

$v=\underset{\mathrm{\Delta }t\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}} \frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}=\frac{ds}{dt}$

Dimension of Instantaneous Speed $\left[L^1{T}^{-1}\right]$

Unit of Instantaneous Speed: $\mathrm{m}/\mathrm{s}\left(\mathrm{S}\mathrm{I}\right),\mathrm{c}\mathrm{m}/\mathrm{s}\left(\mathrm{C}\mathrm{G}\mathrm{S}\right)$

Here is a quick comparison table on instantaneous velocity vs instantaneous speed. 

Here are some additional points on the relationship between instantaneous velocity and speed to keep in mind as you move to preparing for competitive tests, post your CBSE boards. 

• Instantaneous velocity's direction aligns with the tangent to the path. This is how you define projectile motion, too.
• Instantaneous speed remains constant when the velocity's direction changes without magnitude.
• Both instantaneous velocity and speed have identical units and dimensions.
• Correlating the graphical representation of instantaneous velocity and speed - the slope of the position-time graph at a point shows the instantaneous velocity as v=tan⁡, where is the angle with the time axis. Then, instantaneous speed is the magnitude of this slope. For a curved position-time graph, the tangent at a specific time yields the velocity. A horizontal position-time graph indicates zero velocity and speed. Whereas, a negative slope tells us that velocity is in the opposite direction, but speed remains positive.
• A particle in uniform circular motion has constant instantaneous speed but varying instantaneous velocity due to directional changes.
• In one-dimensional motion, instantaneous speed is the absolute value of instantaneous velocity.
• If position $x\left(t\right)=A_0-A_{1}t+A_2{t}^2$ , then velocity $v=\frac{dx}{dt}=-A_1+2A_{2}t$ , and speed is $\left|v\right|$ .
• For constant velocity, instantaneous and average velocities are equal.