# 4.2 Kinetic energy and the work-energy theorem  (Page 3/6)

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Solution for (a)

The net force is the push force minus friction, or ${F}_{\text{net}}\text{= 120 N – 5}\text{.}\text{00 N = 115 N}$ . Thus the net work is

$\begin{array}{lll}{W}_{\text{net}}& =& {F}_{\text{net}}d=\left(\text{115 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{92.0 N}\cdot m=\text{92.0 J.}\end{array}$

Discussion for (a)

This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force.

Strategy and Concept for (b)

The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work.

Solution for (b)

The applied force does work.

${W}_{\mathrm{a}\mathrm{p}\mathrm{p}}={F}_{\mathrm{a}\mathrm{p}\mathrm{p}}d=\left(120\phantom{\rule{.1667em}{0ex}}\mathrm{N}\right)\left(0.800\phantom{\rule{.1667em}{0ex}}\mathrm{m}\right)=96.0\phantom{\rule{.1667em}{0ex}}\mathrm{J}$

The friction force and displacement are in opposite directions. The work done by friction is therefore negative.

${W}_{\mathrm{f}\mathrm{r}}=-{F}_{\mathrm{f}\mathrm{r}}d=-\left(5.00\phantom{\rule{.1667em}{0ex}}\mathrm{N}\right)\left(0.800\phantom{\rule{.1667em}{0ex}}\mathrm{m}\right)=-4.00\phantom{\rule{.1667em}{0ex}}\mathrm{J}$

So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively,

$\begin{array}{lll}{W}_{\text{gr}}& =& 0,\\ {W}_{N}& =& 0,\\ {W}_{\text{app}}& =& \text{96.0 J,}\\ {W}_{\text{fr}}& =& -\text{4.00 J.}\end{array}$

The total work done as the sum of the work done by each force is then seen to be

${W}_{\text{total}}={W}_{\text{gr}}+{W}_{N}+{W}_{\text{app}}+{W}_{\text{fr}}=\text{92}\text{.0 J}.$

Discussion for (b)

The calculated total work ${W}_{\text{total}}$ as the sum of the work by each force agrees, as expected, with the work ${W}_{\text{net}}$ done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach.

## Determining speed from work and energy

Find the speed of the package in [link] at the end of the push, using work and energy concepts.

Strategy

Here the work-energy theorem can be used, because we have just calculated the net work, ${W}_{\text{net}}$ , and the initial kinetic energy, $\frac{1}{2}{m{v}_{0}}^{2}$ . These calculations allow us to find the final kinetic energy, $\frac{1}{2}{\text{mv}}^{2}$ , and thus the final speed $v$ .

Solution

The work-energy theorem in equation form is

${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Solving for $\frac{1}{2}{\text{mv}}^{2}$ gives

$\frac{1}{2}{\text{mv}}^{\text{2}}={W}_{\text{net}}+\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Thus,

$\frac{1}{2}{\text{mv}}^{2}=\text{92}\text{.}0 J+3\text{.}\text{75 J}=\text{95.}\text{75 J.}$

Solving for the final speed as requested and entering known values gives

$\begin{array}{lll}v& =& \sqrt{\frac{2\text{(95.75 J)}}{m}}=\sqrt{\frac{\text{191.5 kg}\cdot {m}^{2}{\text{/s}}^{2}}{\text{30.0 kg}}}\\ & =& \text{2.53 m/s.}\end{array}$

Discussion

Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package.

## 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 can be expressed as the force of friction times the distance traveled and as the change in kinetic energy. Equating both expressions for work gives us a way of finding the distance traveled.

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. To reduce the kinetic energy of the package to zero, the work ${W}_{\text{fr}}$ by friction must be equal to the change in kinetic energy.

${W}_{\mathrm{f}\mathrm{r}}=-{F}_{\mathrm{f}\mathrm{r}}d=\mathrm{\Delta }\mathrm{K}\mathrm{E}$
$-{F}_{\mathrm{f}\mathrm{r}}d=\mathrm{\Delta }\mathrm{K}\mathrm{E}$

and so

$-{F}_{\mathrm{f}\mathrm{r}}d=\frac{1}{2}m{v}^{2}-\frac{1}{2}m{v}_{0}^{2}$

Initial speed is the speed at the instant the push stops (the result determined in the previous example). The final speed is zero for this case and leads to the expression below.

${F}_{\mathrm{f}\mathrm{r}}d=\frac{1}{2}m{v}_{0}^{2}$

Solving for distance, we obtain the result shown below.

$d=\frac{95.75\phantom{\rule{.1667em}{0ex}}\mathrm{J}}{5.00\phantom{\rule{.1667em}{0ex}}\mathrm{N}}=19.2\phantom{\rule{.1667em}{0ex}}\mathrm{m}$

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, ${W}_{\text{net}}=\mathrm{\Delta }\mathrm{K}\mathrm{E}=\frac{1}{2}m{v}^{2}-\frac{1}{2}m{v}_{0}^{2}.$

## 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 30.0 m/s with that of an 80.0-kg astronaut in orbit moving at 7,500 m/s.

$1/\text{250}$

(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.

(a) Calculate the force needed to bring a 950-kg car to rest from a speed of 25.0 m/s 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 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

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