# 7.2 Kinetic energy and the work-energy theorem  (Page 4/7)

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## Work and energy can reveal distance, too

How far does the package in [link] coast after the push, assuming friction remains constant? Use work and energy considerations.

Strategy

We know that once the person stops pushing, friction will bring the package to rest. In terms of energy, friction does negative work until it has removed all of the package’s kinetic energy. The work done by friction is the force of friction times the distance traveled times the cosine of the angle between the friction force and displacement; hence, this gives us a way of finding the distance traveled after the person stops pushing.

Solution

The normal force and force of gravity cancel in calculating the net force. The horizontal friction force is then the net force, and it acts opposite to the displacement, so $\theta =\text{180º}$ . To reduce the kinetic energy of the package to zero, the work ${W}_{\text{fr}}$ by friction must be minus the kinetic energy that the package started with plus what the package accumulated due to the pushing. Thus ${W}_{\text{fr}}=-\text{95}\text{.}\text{75 J}$ . Furthermore, ${W}_{\text{fr}}=fd\prime \phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \phantom{\rule{0.25em}{0ex}}\text{= –}fd\prime$ , where $d\prime$ is the distance it takes to stop. Thus,

$d\prime =-\frac{{W}_{\text{fr}}}{f}=-\frac{-\text{95.75 J}}{\text{5.00 N}},$

and so

$d\prime =\text{19}\text{.2 m}.$

Discussion

This is a reasonable distance for a package to coast on a relatively friction-free conveyor system. Note that the work done by friction is negative (the force is in the opposite direction of motion), so it removes the kinetic energy.

Some of the examples in this section can be solved without considering energy, but at the expense of missing out on gaining insights about what work and energy are doing in this situation. On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone.

## Section summary

• The net work ${W}_{\text{net}}$ is the work done by the net force acting on an object.
• Work done on an object transfers energy to the object.
• The translational kinetic energy of an object of mass $m$ moving at speed $v$ is $\text{KE}=\frac{1}{2}{\text{mv}}^{2}$ .
• The work-energy theorem states that the net work ${W}_{\text{net}}$ on a system changes its kinetic energy, ${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}$ .

## Conceptual questions

The person in [link] does work on the lawn mower. Under what conditions would the mower gain energy? Under what conditions would it lose energy?

Work done on a system puts energy into it. Work done by a system removes energy from it. Give an example for each statement.

When solving for speed in [link] , we kept only the positive root. Why?

## Problems&Exercises

Compare the kinetic energy of a 20,000-kg truck moving at 110 km/h with that of an 80.0-kg astronaut in orbit moving at 27,500 km/h.

$1/\text{250}$

(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.

Confirm the value given for the kinetic energy of an aircraft carrier in [link] . You will need to look up the definition of a nautical mile (1 knot = 1 nautical mile/h).

$1\text{.}1×{\text{10}}^{\text{10}}\phantom{\rule{0.20em}{0ex}}\text{J}$

(a) Calculate the force needed to bring a 950-kg car to rest from a speed of 90.0 km/h in a distance of 120 m (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a).

A car’s bumper is designed to withstand a 4.0-km/h (1.1-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1 m/s.

$2\text{.}8×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N}$

Boxing gloves are padded to lessen the force of a blow. (a) Calculate the force exerted by a boxing glove on an opponent’s face, if the glove and face compress 7.50 cm during a blow in which the 7.00-kg arm and glove are brought to rest from an initial speed of 10.0 m/s. (b) Calculate the force exerted by an identical blow in the gory old days when no gloves were used and the knuckles and face would compress only 2.00 cm. (c) Discuss the magnitude of the force with glove on. Does it seem high enough to cause damage even though it is lower than the force with no glove?

Using energy considerations, calculate the average force a 60.0-kg sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him.

102 N

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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