Ch. 02 - Describing Motion: Kinematics in One Dimension

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 02 - Describing Motion: Kinematics in One Dimension

Problem 30c

Chapter 2, Problem 30c

Figure 2–42 shows the velocity of a train as a function of time. During what periods, if any, was the acceleration constant?

Verified step by step guidance

Examine the velocity vs. time graph provided in the problem. Recall that acceleration is the rate of change of velocity with respect to time, mathematically expressed as \( a = \frac{\Delta v}{\Delta t} \). Constant acceleration implies that the graph should show a straight line (linear relationship) in the velocity vs. time plot.

Identify the segments of the graph where the slope (\( \frac{\Delta v}{\Delta t} \)) is constant. A constant slope indicates that the acceleration is uniform during that time interval.

For each segment of the graph, calculate the slope by selecting two points on the line and using the formula \( \text{slope} = \frac{v_2 - v_1}{t_2 - t_1} \). If the slope remains the same across the segment, the acceleration is constant.

Mark the time intervals corresponding to the segments with constant slope. These intervals represent the periods during which the acceleration was constant.

Summarize the identified time intervals and ensure they align with the graph's visual representation. Clearly state the periods of constant acceleration based on the analysis of the graph.

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Key Concepts

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

Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. It can be constant or variable, depending on whether the velocity changes uniformly or not. In a velocity vs. time graph, constant acceleration is indicated by a straight line, while a curved line suggests changing acceleration.

Velocity-Time Graphs

A velocity-time graph visually represents an object's velocity over time. The slope of the graph indicates acceleration; a positive slope means acceleration, while a negative slope indicates deceleration. Flat sections of the graph indicate constant velocity, and the shape of the graph helps identify periods of constant acceleration.

Intervals of Constant Acceleration

Intervals of constant acceleration occur when the velocity changes at a uniform rate over time. In a velocity-time graph, these intervals are represented by straight line segments. Identifying these intervals involves looking for sections where the slope remains unchanged, indicating that the acceleration is consistent throughout that period.

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Textbook Question

Figure 2–42 shows the velocity of a train as a function of time. At what time was its velocity greatest?

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Textbook Question

A baseball pitcher throws a baseball with a speed of 43 m/s. Estimate the average acceleration of the ball during the throwing motion. In throwing the baseball, the pitcher accelerates it through a displacement of about 3.5 m, from behind the body to the point where it is released (Fig. 2–44).

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Textbook Question

For an object falling freely from rest, show that the distance traveled during each successive second increases in the ratio of successive odd integers (1, 3, 5, etc.). (This was first shown by Galileo.) See Figs. 2–27 and 2–30.

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Textbook Question

The position of an object is given by 𝓍 = At + Bt², where 𝓍 is in meters and t is in seconds. What are the units of A and B?

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Textbook Question

Figure 2–42 shows the velocity of a train as a function of time. During what periods, if any, was the velocity constant?

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Textbook Question

A car traveling 85 km/h slows down at a constant 0.50 m/s² just by 'letting up on the gas.' Calculate the distance it travels during the first and fifth seconds.

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