FIGURE EX2.9 shows the velocity graph of a particle. Draw the particle's acceleration graph for the interval 0 s ≤ t ≤ 4 s.

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Step 1: Understand the relationship between velocity and acceleration. Acceleration is the rate of change of velocity with respect to time, mathematically expressed as \( a = \frac{dv}{dt} \). This means the slope of the velocity-time graph represents the acceleration.

Step 2: Analyze the velocity graph provided. The graph has three distinct segments: (1) from \( t = 0 \) to \( t = 5 \), the velocity increases linearly, (2) from \( t = 5 \) to \( t = 7 \), the velocity decreases linearly, and (3) from \( t = 7 \) to \( t = 10 \), the velocity remains constant.

Step 3: Calculate the slope for each segment to determine the acceleration. For \( t = 0 \) to \( t = 5 \), the slope is \( \frac{\Delta v}{\Delta t} = \frac{8 - 0}{5 - 0} = 1.6 \, \text{m/s}^2 \). For \( t = 5 \) to \( t = 7 \), the slope is \( \frac{\Delta v}{\Delta t} = \frac{8 - 6}{7 - 5} = -1 \, \text{m/s}^2 \). For \( t = 7 \) to \( t = 10 \), the slope is \( \frac{\Delta v}{\Delta t} = \frac{6 - 6}{10 - 7} = 0 \, \text{m/s}^2 \).

Step 4: Draw the acceleration graph based on the calculated slopes. For \( t = 0 \) to \( t = 5 \), plot a horizontal line at \( a = 1.6 \). For \( t = 5 \) to \( t = 7 \), plot a horizontal line at \( a = -1 \). For \( t = 7 \) to \( t = 10 \), plot a horizontal line at \( a = 0 \).

Step 5: Label the axes of the acceleration graph. The x-axis represents time \( t \) in seconds, and the y-axis represents acceleration \( a \) in \( \text{m/s}^2 \). Ensure the graph reflects the calculated values for each interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction, typically expressed in meters per second (m/s). In the context of the provided graph, the velocity of the particle varies over time, indicating how fast and in which direction the particle is moving at any given moment.

Acceleration

Acceleration is the rate of change of velocity over time, measured in meters per second squared (m/s²). It indicates how quickly an object is speeding up or slowing down. To derive the acceleration graph from the velocity graph, one must analyze the slope of the velocity curve; a positive slope indicates positive acceleration, while a negative slope indicates deceleration.

Graph Interpretation

Interpreting graphs is crucial in physics as it allows for the visualization of relationships between variables. In this case, the velocity-time graph provides insights into the motion of the particle, such as periods of constant velocity, acceleration, and deceleration. Understanding how to read and analyze these graphs is essential for accurately determining the corresponding acceleration graph.