Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 30c

Chapter 2, Problem 30c

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). What distance does the cat move during the first $4.5$ s? From $t equals 0$ to $t equals 7.5$ s?

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To find the distance the cat moves, we need to calculate the area under the velocity-time graph. The area under the graph represents the displacement of the cat.

For the first graph, observe that the velocity changes linearly from -3 m/s to 3 m/s over 7.5 seconds. This forms a trapezoid from t = 0 to t = 7.5 s.

Calculate the area of the trapezoid from t = 0 to t = 4.5 s. The trapezoid has a base from t = 0 to t = 4.5 s, a height of -3 m/s at t = 0, and a height of 0 m/s at t = 4.5 s.

Calculate the area of the trapezoid from t = 0 to t = 7.5 s. The trapezoid has a base from t = 0 to t = 7.5 s, a height of -3 m/s at t = 0, and a height of 3 m/s at t = 7.5 s.

For the second graph, observe that the velocity changes linearly from 8 cm/s to 0 cm/s over 7 seconds. This forms a triangle. Calculate the area of the triangle from t = 0 to t = 7 s, which represents the displacement in cm.

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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, indicating how fast an object is moving and in which direction. In this scenario, the cat's velocity is represented on the y-axis of the graph, showing how its speed changes over time.

Area under the Velocity-Time Graph

The area under a velocity-time graph represents the displacement of the object over a given time interval. For a linear graph, this area can be calculated using geometric shapes, such as triangles or rectangles. In this case, calculating the area under the curve from 0 to 4.5 seconds will provide the distance the cat has moved during that time.

Linear Motion

Linear motion refers to the movement of an object along a straight path. In this question, the cat is moving along the x-axis, and its velocity decreases linearly over time, indicating a uniform deceleration. Understanding linear motion is crucial for analyzing the cat's movement and calculating the distance traveled based on the velocity-time graph.

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Related Practice

Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). What is the cat's acceleration at $t equals 3.0$ s? At $t equals 6.0$ s? At $t equals 7.0$ s?

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Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). Find the cat's velocity at $t equals 4.0$ s and at $t equals 7.0$ s.

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Textbook Question

A small block has constant acceleration as it slides down a frictionless incline. The block is released from rest at the top of the incline, and its speed after it has traveled $6.80$ m to the bottom of the incline is $3.80$ m/s. What is the speed of the block when it is $3.40$ m from the top of the incline?

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Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). Assuming that the cat started at the origin, sketch clear graphs of the cat's acceleration and position as functions of time.

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Textbook Question

At launch a rocket ship weighs $4.5$ million pounds. When it is launched from rest, it takes $8.00$ s to reach $161$ km/h; at the end of the first $1.00$ min, its speed is $1610$ km/h. What is the average acceleration (in m/s²) of the rocket (i) during the first $8.00$ s and (ii) between $8.00$ s and the end of the first $1.00$ min?

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Textbook Question

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of $2.80$ m/s². At the same instant a truck, traveling with a constant speed of $20.0$ m/s, overtakes and passes the car. How far beyond its starting point does the car overtake the truck?

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