Ch. 02 - Describing Motion: Kinematics in One Dimension

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 02 - Describing Motion: Kinematics in One Dimension

Problem 27a

Chapter 2, Problem 27a

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?

Verified step by step guidance

Step 1: Start by analyzing the given equation for position: 𝓍 = At + Bt². Here, 𝓍 represents position in meters (m), t represents time in seconds (s), and A and B are constants with unknown units.

Step 2: Focus on the term At. Since 𝓍 is in meters, the product At must also have units of meters. The unit of t is seconds (s), so the unit of A must be meters per second (m/s) to ensure the product At has units of meters.

Step 3: Now, consider the term Bt². Again, since 𝓍 is in meters, the product Bt² must also have units of meters. The unit of t² is seconds squared (s²), so the unit of B must be meters per second squared (m/s²) to ensure the product Bt² has units of meters.

Step 4: Summarize the findings: The unit of A is meters per second (m/s), and the unit of B is meters per second squared (m/s²).

Step 5: Verify the consistency of units in the equation: Substitute the derived units of A and B into the equation 𝓍 = At + Bt². Confirm that both terms At and Bt² have units of meters, ensuring the equation is dimensionally consistent.

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Key Concepts

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

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this context, the equation 𝓍 = At + Bt² represents the position of an object as a function of time, where A and B are coefficients that influence the object's motion. Understanding these equations is essential for analyzing how position changes with time.

Dimensional Analysis

Dimensional analysis is a method used to convert units and check the consistency of equations. By analyzing the dimensions of each term in the equation, we can determine the units of A and B. This technique ensures that all terms in the equation are compatible, which is crucial for solving physics problems.

Units of Measurement

Units of measurement provide a standard for quantifying physical quantities. In this equation, 𝓍 is measured in meters (m) and time t in seconds (s). The units of A and B can be derived from the equation by ensuring that the terms on both sides have the same dimensions, leading to a clear understanding of their physical significance.

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Related Practice

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

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

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

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Textbook Question

The position of an object along a straight tunnel as a function of time is plotted in Fig. 2–40. What is its average velocity between t = 25.0 s and t = 30.0 s?

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Textbook Question

A bowling ball traveling with constant speed hits the pins at the end of a bowling lane 16.5 m long. The bowler hears the sound of the ball hitting the pins 2.75 s after the ball is released from his hands. What is the speed of the ball, assuming the speed of sound is 340 m/s?

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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 acceleration constant?

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