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 28b

Chapter 2, Problem 28b

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?

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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 pilot who accelerates at more than $4 g$ begins to 'gray out' but doesn't completely lose consciousness. Assuming constant acceleration, what is the shortest time that a jet pilot starting from rest can take to reach Mach $4$ (four times the speed of sound) without graying out? (Use $331$ m/s for the speed of sound in cold air.)

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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 pilot who accelerates at more than $4 g$ begins to 'gray out' but doesn't completely lose consciousness. How far would the plane travel during this period of acceleration? (Use $331$ m/s for the speed of sound in cold air.)

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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

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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