Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 24b

Chapter 2, Problem 24b

A car sits on an entrance ramp to a freeway, waiting for a break in the traffic. Then the driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of $20$ m/s ( $45$ mi/h) when it reaches the end of the $120$ -m-long ramp. How much time does it take the car to travel the length of the ramp?

Verified step by step guidance

Identify the known values: initial velocity $ v_0 = 0 $ m/s (since the car starts from rest), final velocity $ v = 20 $ m/s, and displacement $ s = 120 $ m.

Use the kinematic equation that relates velocity, acceleration, and displacement: $ v^2 = v_0^2 + 2as $. Since $ v_0 = 0 $, this simplifies to $ v^2 = 2as $.

Rearrange the equation to solve for acceleration $ a $: $ a = \frac{v^2}{2s} $. Substitute the known values to find $ a $.

Use the kinematic equation $ v = v_0 + at $ to solve for time $ t $. Since $ v_0 = 0 $, it simplifies to $ t = \frac{v}{a} $.

Substitute the values of $ v $ and $ a $ into the equation $ t = \frac{v}{a} $ to find the time it takes for the car to travel the length of the ramp.

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Key Concepts

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

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In this problem, the car accelerates uniformly, meaning its acceleration remains the same throughout its motion along the ramp. This allows us to use kinematic equations to determine the time taken to travel a specific distance.

Kinematic Equations

Kinematic equations are mathematical formulas used to describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For this problem, the equation v^2 = u^2 + 2as can be used, where v is final velocity, u is initial velocity, a is acceleration, and s is displacement, to find the time taken.

Initial Conditions

Initial conditions are the starting values of variables in a physics problem, such as initial velocity and position. Here, the car starts from rest, meaning its initial velocity is zero. This simplifies calculations, as the initial velocity term in kinematic equations becomes zero, allowing us to focus on the relationship between acceleration, final velocity, and displacement.

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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 $2.30$ ). 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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Textbook Question

A car sits on an entrance ramp to a freeway, waiting for a break in the traffic. Then the driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of $20$ m/s ( $45$ mi/h) when it reaches the end of the $120$ -m-long ramp. What is the acceleration of the car?

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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 $2.30$ ). 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

The fastest measured pitched baseball left the pitcher's hand at a speed of $45.0$ m/s. If the pitcher was in contact with the ball over a distance of $1.50$ m and produced constant acceleration, what acceleration did he give the ball?

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Textbook Question

The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than $250$ m/s². If you are in an automobile accident with an initial speed of $105$ km/h ( $65$ mi/h) and are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

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