Motion in a Straight Line | Class 11 Physics Notes

2.2 Instantaneous velocity and speed

Average velocity over a time interval gives the overall rate of change of position but does not describe how velocity varies at each instant within that interval. To capture this, instantaneous velocity is defined as the velocity of an object at a specific instant of time.

Mathematically, instantaneous velocity v at time t is defined as the limit of the average velocity as the time interval Δt approaches zero:

v = lim (Δt → 0) (Δx / Δt) = dx/dt

Here, dx/dt is the derivative of position x with respect to time t, representing the rate of change of position at that instant. This definition uses calculus to find the exact velocity at any moment.

Graphically, instantaneous velocity at time t can be found as the slope of the tangent to the position-time graph at that point. For example, in Fig. 2.1, the velocity at t = 4 s is the slope of the tangent drawn at that instant on the position-time curve.

Numerically, one can calculate average velocities over smaller and smaller intervals centered at t = 4 s, as shown in Table 2.1. As Δt decreases from 2 s to 0.01 s, the average velocity approaches a limiting value of approximately 3.84 m/s, which is the instantaneous velocity at t = 4 s.

If the position as a function of time is known explicitly, differentiation can be used directly to find instantaneous velocity. For example, if x = a + b t², then v = dx/dt = 2 b t.

Instantaneous speed is the magnitude of instantaneous velocity. For example, velocities of +24 m/s and -24 m/s both correspond to a speed of 24 m/s. While average speed over a finite interval is always greater than or equal to the magnitude of average velocity, instantaneous speed equals the magnitude of instantaneous velocity at any instant.

📊 Diagram: Fig. 2.1 Determining velocity from position-time graph. Velocity at t = 4 s is the slope of the tangent to the graph at that instant.

🔗 Connection: This section leads to the next by introducing velocity changes over time, which motivates the concept of acceleration.

Table on page 2 (6×7)

Table on page 10 (3×5)

| Displacement | Δx | [L] | m | = x_{0} - x_{1}

Table on page 10 (2×5)

| (b) Instantaneous | v | | | = \frac{\lim}{\Delta t} \cdot \frac{\Delta x}{\Delta t} = \frac{\mathrm{d}x}{\mathrm{d}t}

Table on page 11 (1×5)

| Acceleration