Sketch the v vs. t graph for the object whose displacement as a function of time is given by Fig. 2–40.

Verified step by step guidance

1. Analyze the displacement vs. time graph (Fig. 2–40) to determine the slope of the graph at different time intervals. The slope of a displacement vs. time graph represents the velocity of the object.
2. Identify the regions of the graph where the slope is constant, increasing, decreasing, or zero. A constant slope indicates constant velocity, an increasing slope indicates acceleration, a decreasing slope indicates deceleration, and a zero slope indicates the object is at rest.
3. For each time interval, calculate the slope of the displacement vs. time graph. Use the formula for slope: \( \text{slope} = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in displacement and \( \Delta x \) is the change in time.
4. Plot the calculated slopes (velocities) on a velocity vs. time graph. Ensure that the velocity values correspond to the correct time intervals from the displacement vs. time graph.
5. Connect the points on the velocity vs. time graph appropriately. If the slope of the displacement graph changes linearly, the velocity graph will show a straight line. If the slope changes abruptly, the velocity graph will have a sharp transition.

Key Concepts

Displacement vs. Time Relationship

Displacement as a function of time describes how the position of an object changes over time. The slope of the displacement vs. time graph indicates the object's velocity. A steeper slope represents a higher velocity, while a flat line indicates the object is at rest. Understanding this relationship is crucial for interpreting how displacement affects velocity.

Velocity vs. Time Graph

The velocity vs. time graph represents how an object's velocity changes over time. The area under the velocity curve corresponds to the displacement of the object. If the velocity is constant, the graph will be a horizontal line, while varying velocity will create a curve. This graph is essential for visualizing motion and understanding acceleration.

Graphical Interpretation of Motion

Graphical interpretation involves analyzing graphs to extract information about an object's motion. By sketching the velocity vs. time graph based on the displacement graph, one can identify key features such as acceleration, deceleration, and constant velocity. This skill is vital for solving problems in kinematics and understanding the dynamics of moving objects.