NCERT Physics Class 11 Chapter 2
2.2 Instantaneous Velocity and Speed
“The average velocity tells us how fast an object has been moving over a given time interval...”
Easy Meaning
- Average velocity gives motion over a long interval of time.
- It does NOT tell the exact velocity at a particular moment.
Example
If a car travels:
- 100 m in 10 s
vavg =
Δx / Δt
= 100 / 10 = 10 m/s
= 100 / 10 = 10 m/s
But the car may move:
- slowly at first
- faster later
So average velocity hides instant changes.
“...but does not tell us how fast it moves at different instants of time...”
Easy Meaning
- We need velocity at one exact moment.
- Example:
- velocity exactly at 4 s
- not from 0 to 10 s
Instantaneous Velocity
“For this, we define instantaneous velocity or simply velocity v at an instant t.”
Definition
Instantaneous velocity = velocity at a particular instant of time.
Example
- Speedometer of a bike shows instantaneous speed.
Mathematical Definition
“The velocity at an instant is defined as the limit of average velocity...”
Idea
Take smaller and smaller time intervals.
- When interval becomes extremely tiny:
- average velocity becomes instantaneous velocity.
Formula of Instantaneous Velocity
v = limΔt→0 (Δx / Δt)
Meaning of Symbols
- (v) → instantaneous velocity
- (Δx) → small displacement
- (Δt) → small time interval
Calculus Form
v = dx / dt
Meaning
Velocity = rate of change of position with time.
“...the quantity on the right hand side is the differential coefficient of x with respect to t...”
Easy Meaning
In calculus:
dx / dt
means how quickly position changes with time.
“It is the rate of change of position with respect to time...”
Important Point
- Velocity tells:
- how fast position changes
- and in which direction
Finding Velocity Graphically
“We can use Eq. (2.1a) for obtaining the value of velocity graphically or numerically.”
Two Methods
- Graph method
- Numerical/table method
“Suppose we want to obtain graphically the value of velocity at t = 4 s...”
Meaning
We want exact velocity at 4 seconds.
“Let us take Δt = 2 s centered at t = 4 s.”
Meaning
Choose time interval around 4 s:
- from 3 s to 5 s
Average Velocity from Graph
Slope of Line
Slope = Δx / Δt
Meaning
Slope of position-time graph gives velocity.
“The slope of line P₁P₂ gives average velocity over interval 3 s to 5 s.”
Easy Meaning
- Draw line between two points.
- Measure rise/change in position.
- Divide by time interval.
“Now, we decrease the value of Δt from 2 s to 1 s.”
Meaning
Take smaller intervals:
- 3.5 to 4.5
- 3.75 to 4.25
- etc.
“In the limit Δt → 0, the line P₁P₂ becomes tangent to the curve...”
Very Important Concept
- When interval becomes extremely small:
- secant line changes into tangent.
Tangent Slope = Instantaneous Velocity
Tangent Concept
“Velocity at t = 4 s is given by the slope of tangent at point P.”
Final Conclusion
Instantaneous velocity at a point = slope of tangent to position-time graph at that point.
Numerical Method
“It is difficult to show this process graphically...”
Meaning
Exact tangent drawing is hard. So tables and calculations are easier.
“Table 2.1 gives values of Δx/Δt...”
What Table Shows
As:
Δt → 0
The value of:
Δx / Δt
approaches a fixed number.
Observation from Table
| Δt gets smaller | Velocity value |
|---|---|
| 2.0 | 3.92 |
| 1.0 | 3.86 |
| 0.5 | 3.845 |
| 0.1 | 3.8402 |
| 0.01 | 3.8400 |
Final Instantaneous Velocity
v ≈ 3.84 m/s
Important Concepts Summary
Average Velocity
vavg = Δx / Δt
- For large interval
Instantaneous Velocity
v = dx / dt
- Velocity at exact instant
Graph Rule
Position-Time Graph
- Slope of secant → average velocity
- Slope of tangent → instantaneous velocity
Easy Real-Life Examples
| Situation | Type |
|---|---|
| Average speed of trip | Average velocity |
| Car speedometer reading | Instantaneous velocity |
| Straight line slope | Velocity from graph |
One-Line Summary:
Instantaneous velocity is the velocity of an object at one exact moment and is equal to the slope of the tangent to the position-time graph at that instant.
Learn Instantaneous Velocity and Speed Class 11 Physics with formulas, graphs, examples, numerical methods, and NCERT explanations.
Internal Links
Add these naturally throughout the article:
Motion & Kinematics
Motion in a Straight Line Notes
Distance and Displacement Explained
Speed vs Velocity Difference
Average Velocity Formula and Examples
Acceleration in One Dimension
Position-Time Graph Explained
Velocity-Time Graph Explained
Equations of Motion Class 11
Calculus & Concepts
Introduction to Differentiation in Physics
Graphical Interpretation of Motion
Slope of a Graph in Physics
Numerical Methods in Kinematics
Exam Preparation
Class 11 Physics Important Questions
NEET Kinematics Questions
JEE Motion in a Straight Line Problems
NCERT Solutions Motion in a Straight Line
Previous Year Physics Questions
 |
| Instantaneous velocity is equal to the slope of the tangent to the position-time graph at a given instant. |
No comments:
Post a Comment