Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 73a

Chapter 2, Problem 73a

When a 1984 Alfa Romeo Spider sports car accelerates at the maximum possible rate, its motion during the first 20 s is extremely well modeled by the simple equation v_x² = (2P/m)t, where P = 3.6 ✕ 10⁴ watts is the car's power output, m = 1200 kg is its mass, and v_x is in m/s. That is, the square of the car's velocity increases linearly with time. Find an algebraic expression in terms of P, m, and t for the car's acceleration at time t.

Verified step by step guidance

Start with the given equation for the square of the car's velocity:

v_x^2 = \(\frac{2Pt}{m}\)

, where

P

is the power output,

m

is the mass, and

t

is time.

To find the car's acceleration, recall that acceleration is the time derivative of velocity:

a(t) = \(\frac{dv_x}{dt}\)

Differentiate both sides of the velocity equation with respect to time. First, rewrite the velocity as

v_x = \(\sqrt{\frac{2Pt}{m}\)}

. Then, apply the chain rule to differentiate:

\(\frac{dv_x}{dt}\) = \(\frac{1}{2}\) \(\cdot\) \(\left\)(\(\frac{2Pt}{m}\[\right\))^{-1/2} \(\cdot\) \(\frac{d}{dt}\]\left\)(\(\frac{2Pt}{m}\)\(\right\))

Simplify the derivative. The term

\(\frac{d}{dt}\[\left\)(\(\frac{2Pt}{m}\]\right\))

simplifies to

\(\frac{2P}{m}\)

, so the acceleration becomes:

a(t) = \(\frac{1}{2}\) \(\cdot\) \(\left\)(\(\frac{2Pt}{m}\)\(\right\))^{-1/2} \(\cdot\) \(\frac{2P}{m}\)

Combine and simplify the terms to express the acceleration in terms of

P

m

, and

t

a(t) = \(\frac{P}{m}\) \(\cdot\) \(\left\)(\(\frac{m}{2Pt}\)\(\right\))^{1/2}

. This is the algebraic expression for the car's acceleration at time

t

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Key Concepts

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

Power and Work

Power is defined as the rate at which work is done or energy is transferred over time. In the context of a car, the power output indicates how quickly the engine can convert fuel into kinetic energy, affecting the car's acceleration. The relationship between power, force, and velocity is crucial for understanding how a car accelerates, as it directly influences the car's ability to increase its speed.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is fundamental in analyzing the motion of the Alfa Romeo Spider, as it allows us to relate the force generated by the car's engine (derived from its power output) to its acceleration. Understanding this law is essential for deriving the expression for acceleration in the given scenario.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this case, the problem states that the square of the car's velocity increases linearly with time, suggesting a specific relationship between velocity, acceleration, and time. These equations will help in deriving the algebraic expression for acceleration by relating the variables of power, mass, and time.

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Related Practice

Textbook Question

A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

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Textbook Question

David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s² at the instant when David passes. What is her speed as she passes him?

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Textbook Question

A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_x = a₀ ─ kv_x, where a₀ is the initial acceleration and k is a constant. Find an expression for k in terms of a₀ and the car's top speed v_max.

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Textbook Question

A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at 2500m/s . It is loaded with 600 kg of fuel, which it burns in 30 s. What is the rocket's speed 10 s, 20 s, and 30 s after launch?

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Textbook Question

David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s² at the instant when David passes. How far does Tina drive before passing David?

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Textbook Question

If a Tesla Model S P100D in 'Ludicrous mode' is pushed to its limit, the first $3.0 s$ of acceleration can be modeled as

$a_{x} = {\begin{matrix} (35 {m/s}^{3}) t & 0 s \leq t \leq 0.40 s \\ 14.6 {m/s}^{2} - (1.5 {m/s}^{3}) t & 0.40 s \leq t \leq 3.0 s \end{matrix}$

What acceleration would be needed to achieve the same speed in the same time at constant acceleration? Give your answer as a multiple of $g$ .

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