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 28a

Chapter 2, Problem 28a

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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Identify the known values: initial velocity (v₀) = 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 these quantities: v² = v₀² + 2as, where v is the final velocity, v₀ is the initial velocity, a is the acceleration, and s is the displacement.

Substitute the known values into the equation: (20 m/s)² = (0 m/s)² + 2a(120 m).

Simplify the equation to solve for acceleration (a): 400 m²/s² = 240a m.

Rearrange the equation to isolate a: a = 400 m²/s² / 240 m.

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Key Concepts

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

Kinematics Equations

Kinematics equations describe the motion of objects without considering the forces that cause the motion. For constant acceleration, the equation v^2 = u^2 + 2as can be used, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement. This equation helps determine the car's acceleration given its initial and final velocities and the distance traveled.

Constant Acceleration

Constant acceleration implies that the rate of change of velocity is uniform over time. In this scenario, the car's acceleration remains the same throughout its motion on the ramp. Understanding constant acceleration is crucial for applying kinematic equations correctly, as it simplifies calculations by allowing the use of specific formulas designed for uniform acceleration.

Initial Conditions

Initial conditions refer to the state of an object at the beginning of its motion. Here, the car starts from rest, meaning its initial velocity is zero. Recognizing initial conditions is essential for solving physics problems, as they provide the necessary starting point for applying kinematic equations and determining other variables like acceleration or time.

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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. How much time does it take the car to travel the length of the ramp?

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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 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 $2.30$ ). 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 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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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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