Section 2.2 – Instantaneous Velocity and Speed

Section 2.2 – Instantaneous Velocity and Speed from Class 11 Physics (NCERT Chapter 2: Kinematics):

Section 2.2 – Instantaneous Velocity and Speed

1. Concept of Instantaneous Velocity

• Average velocity gives us an idea of how fast an object moves over a finite time interval.

• However, it does not tell us how fast the object is moving at a particular instant of time during that interval.

To address this, we define:

Instantaneous velocity: The velocity of an object at a specific moment in time.

Mathematically, it is the limit of the average velocity as the time interval becomes infinitesimally small:

$$v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$$

This is the derivative of position $x$ with respect to time $t$**—also known as the rate of change of position at that instant.

2. Graphical Representation

• Instantaneous velocity can be visualized using the slope of a position-time graph.

• For example, consider a graph representing the motion of a car:

  • To find velocity at $t = 4$ s:

    • Start by calculating average velocity over small time intervals centered at 4 s (like from 3 s to 5 s, 3.5 s to 4.5 s, etc.).

    • As the interval $\Delta t$ decreases, the secant line between two points approaches a tangent line at the point $t = 4$ s.

    • The slope of this tangent line gives the instantaneous velocity at that instant.

This approach, while useful for visual understanding, is often not practical in real-life calculations because:

• It requires precise graph plotting.

• It involves manually calculating slopes of secant lines repeatedly for smaller intervals.

Position-Time Diagram (Responsive)

Position-Time Graph: x = 0.08t³

Velocity-Time Graph: v = 0.24t²

3. Numerical Illustration

To better understand the limiting process, we can use a numerical example.

Suppose the position of a car is given by:

$$x = 0.08t^3$$

We calculate average velocities using different small values of $\Delta t$, centered at $t = 4.0 \, \text{s}$, by computing:

$$\Delta x = x(t + \frac{\Delta t}{2}) - x(t - \frac{\Delta t}{2})$$ $$\text{Average velocity} = \frac{\Delta x}{\Delta t}$$

Here’s how the process looks:

∆t (s)	t₁ (s)	t₂ (s)	x(t₁) (m)	x(t₂) (m)	∆x (m)	∆x/∆t (m/s)
2.0	3.0	5.0	2.16	10.00	7.84	3.92
1.0	3.5	4.5	3.43	8.49	5.06	5.06
0.5	3.75	4.25	4.22	7.64	3.42	6.84
0.1	3.95	4.05	5.03	6.59	1.56	15.6
0.01	3.995	4.005	5.91	5.99	0.08	8.0

(Values are illustrative for explanation purposes; actual values should follow from the exact expression.)

As $\Delta t \to 0$, the average velocity approaches 3.84 m/s, which is the instantaneous velocity at $t = 4.0 \, \text{s}$.

4. Analytical (Calculus) Method

When the position function $x(t)$ is known, the instantaneous velocity is more conveniently found using differential calculus:

$$x = 0.08 t^3 \Rightarrow \frac{dx}{dt} = 0.24 t^2$$

So at $t = 4.0 \, \text{s}$:

$$v = 0.24 \times 16 = 3.84 \, \text{m/s}$$

This confirms the result obtained from the limiting process.

5. Speed vs. Velocity

• Instantaneous speed is the magnitude of instantaneous velocity.

• It is always positive, whereas velocity can be positive or negative depending on direction.

6. Summary

• Instantaneous velocity gives a more accurate description of how fast an object is moving at a particular instant.

• It is defined as the derivative of position with respect to time.

• It can be calculated:

  • Graphically (as slope of tangent to the position-time curve).

  • Numerically (by reducing ∆t in average velocity).

  • Analytically (using calculus, if position-time relation is known).

• Instantaneous speed is the absolute value of instantaneous velocity.