(a) By taking the slope of the curve in Figure 2.72, verify...

(a) By taking the slope of the curve in Figure $2.72,$ verify that the velocity of the jet car is 115 $\mathrm{m} / \mathrm{s}$ at $t=20 \mathrm{s}$ . (b) By taking the slope of the curve at any point in Figure 2.73 , verify that the jet car's acceleration is 5.0 $\mathrm{m} / \mathrm{s}^{2}$ .

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Key Concepts

Derivative

A derivative represents the rate of change of a quantity with respect to another. In kinematics, taking the derivative of a position versus time function gives the instantaneous velocity, and similarly, differentiating the velocity function provides the instantaneous acceleration.

Instantaneous Velocity

Instantaneous velocity is the rate at which position changes at a specific moment in time. It is determined by calculating the slope of the displacement-time graph at the point of interest, providing a precise measure of how fast an object is moving at that exact moment.

Instantaneous Acceleration

Instantaneous acceleration is defined as the rate of change of velocity at a specific instant. It is obtained by finding the slope of the velocity-time graph at a particular point, which reveals how quickly the velocity of an object is increasing or decreasing.

Graphical Analysis in Kinematics

Graphical analysis involves interpreting motion graphs to extract key kinematic quantities such as velocity and acceleration. By analyzing the slopes of position-time and velocity-time curves, one can verify kinematic values and understand the overall motion of an object.

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