Instantaneous Velocity and Speed

Instantaneous Velocity

Definition: Instantaneous velocity is the velocity of an object at a particular instant of time. It is defined as the limit of average velocity as the time interval approaches zero:

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

• It is a vector quantity.
• It indicates both magnitude and direction at a specific instant.

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Graphical Interpretation

The slope of the tangent to the position-time (x–t) graph at any point gives the instantaneous velocity at that point.

• A positive slope indicates positive velocity (forward direction).
• A negative slope indicates negative velocity (backward direction).
• A zero slope indicates that the object is momentarily at rest.

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Instantaneous Speed

• Defined as the magnitude of instantaneous velocity:

$$\text{Instantaneous Speed} = \left| \frac{dx}{dt} \right|$$

• It is a scalar quantity.
• It is always positive or zero.

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Comparing Speed and Velocity

• Speed shows only how fast
• Velocity shows how fast and in which direction

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Summary

• Instantaneous velocity: Rate of change of position at a moment.
• Speed: Magnitude of velocity.
• Use tangent slope on x–t graph to determine instantaneous velocity.

Example 1:

Position: $x(t) = 4t + 2$ $$\frac{dx}{dt} = 4 ⇒ v = 4\ \text{m/s}$$

Example 2:

Position: $x(t) = 2t^2 + 1$ $$\frac{dx}{dt} = 4t ⇒ v(2) = 8\ \text{m/s}$$

Example 3:

A curved x–t graph is given. Tangent drawn at $t = 3$ s has slope 5. ⇒ Instantaneous velocity = 5 m/s