In Figure 2.11 b, the area under the velocity-time graph...

In Figure $2.11 \mathrm{~b},$ the area under the velocity-time graph and between the vertical axis and time $t$ (vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. (a) Compute their areas. (b) Explain how the sum of the two areas compares with the expression on the right-hand side of Equation $2.16 .$

Key Concepts

Kinematic Equation Correspondence

The sum of the areas of the rectangle and triangle, obtained through geometric decomposition, corresponds directly to a common kinematic equation in uniformly accelerated motion. Typically, the displacement can be expressed as the initial velocity times time plus one half of the acceleration times the square of time. This equivalence demonstrates the connection between graphical analysis and analytical expressions in kinematics.

Displacement

Displacement in kinematics is defined as the net change in position of an object. In the context of a velocity–time graph, it is represented by the area under the curve. This area computes the cumulative effect of velocity over an interval of time, effectively summing up the small amounts of distance traveled during each time period.

Velocity-Time Graph

A velocity–time graph is a plot that shows an object’s velocity as a function of time. The graph provides visual insights into the motion of the object, and areas under the curve correspond to displacements. By analyzing the shape of the graph, one can determine not only the displacement but also the motion characteristics such as acceleration and deceleration.

Geometric Area Calculation

The method of decomposing the area under the curve into simple geometric shapes like rectangles and triangles is a fundamental strategy in physics and calculus. Each shape has a well-known formula for calculating its area—length times width for a rectangle and one half base times height for a triangle. This breakdown simplifies the calculation of displacement when the velocity–time graph is composed of linear segments.

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