Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 9a

Chapter 2, Problem 9a

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

A particle starts from at and moves with the velocity graph shown in FIGURE EX2.6. Does this particle have a turning point? If so, at what time?

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Textbook Question

A particle starts from x₀ = 10 m at t₀= 0 s and moves with the velocity graph shown in FIGURE EX2.6. What is the object’s position at t = 2 s and 4 s?

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Textbook Question

FIGURE EX2.8 is a somewhat idealized graph of the velocity of blood in the ascending aorta during one beat of the heart. Approximately how far, in cm, does the blood move during one beat?

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Textbook Question

FIGURE EX2.8 showed the velocity graph of blood in the aorta. What is the blood's acceleration during each phase of the motion, speeding up and slowing down?

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Textbook Question

FIGURE EX2.12 shows the velocity-versus-time graph for a particle moving along the x-axis. Its initial position is at x₀ = 2 m at t₀ = 0 s. What are the particle's position, velocity, and acceleration at t = 1.0 s?