Ch. 07 - Work and Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 07 - Work and Energy

문제 68

7장, 문제 68

A car traveling at a velocity v can stop in a minimum distance d. What would be the car’s minimum stopping distance if it were traveling at a velocity of 2v?

검증된 단계별 안내

Recognize that the stopping distance of a car is determined by the work-energy principle, which states that the work done by the braking force is equal to the car's initial kinetic energy. The formula for kinetic energy is:

E_{k} = (1/2)mv^2

, where m is the mass of the car and v is its velocity.

The work done by the braking force is given by:

W = Fd

, where F is the braking force and d is the stopping distance. Assuming the braking force F is constant, the stopping distance can be expressed as:

d = (1/2)mv^2 / F

Now, consider the case where the velocity is doubled to

2v

. Substitute

v = 2v

into the stopping distance formula:

d' = (1/2)m(2v)^2 / F

Simplify the expression for

d'

d' = (1/2)m(4v^2) / F = 4((1/2)mv^2 / F)

. Notice that this is four times the original stopping distance

d

Conclude that if the car's velocity is doubled, the minimum stopping distance will increase by a factor of 4, assuming the braking force remains constant.

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주요 개념

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Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and displacement. In this context, understanding how velocity affects stopping distance is crucial, as it relates to the equations of motion that govern an object's behavior when it comes to a stop.

Stopping Distance

Stopping distance is the total distance a vehicle travels from the moment the brakes are applied until it comes to a complete stop. It is influenced by factors such as initial speed, reaction time, and braking force. The relationship between speed and stopping distance is quadratic, meaning that if the speed doubles, the stopping distance increases by a factor of four, which is essential for solving the given problem.

Energy and Work

The concepts of energy and work are fundamental in understanding how a vehicle stops. The kinetic energy of the car, which is proportional to the square of its velocity, must be dissipated through work done by the brakes to bring the car to a stop. This relationship highlights how changes in speed directly affect the energy that needs to be managed, thereby influencing the stopping distance.

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