Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

문제 73a

2장, 문제 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.

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.