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3. Laws Governing Motion

Motion is all around us—people walk, cloudsmove, rain falls, and water flows. Things are movingwherever we look, and motion seems to be associatedwith all the changes we observe. Understanding motionis an important starting point in understanding theworld. Newton helped us to understand motion in termsof the three laws he published in 1687. Although hislaws of motion appear relatively simple, his perspectiveis not intuitively obvious and requires a retraining of theway we think. We will now describe these laws ofmotion and try to understand their meaning, illustratingthem with common experience, but common experienceseen in a new way.

The First Law of Motion

Before dealing with all kinds of motion, we mustfirst ask: How do objects move if they are left alone?What is the “natural” motion of free objects? Onlywhen we know the answers to these questions do weknow what remains to be explained. If free objectsmove in a particular way, objects that move in someother way are not free and their motions must beexplained by another law. The First Law of Motioncorrectly describes the motion of free objects:

Every object continues in its state of rest, or ofuniform motion in a straight line with unchang-ing speed, unless compelled to do otherwise byforces acting upon it.

It seems obvious that an object at rest remains atrest if it is left alone, yet the consequence can some-times be startling. A magician depends on this lawwhen he pulls the tablecloth from a table, leaving thedinner service undisturbed. The plates and goblets areat rest and remain at rest unless the tablecloth “com-pels” them to do otherwise.

A less entertaining manifestation of the law occurswhen a stopped car with passengers is struck frombehind. The passengers’ heads momentarily remain atrest while the car and the rest of their bodies are “com-pelled” to move forward by the force of the impact.This results in stretching and bone dislocation known aswhiplash injury.

This First Law also states that moving objects, if

left to themselves, will continue to move in a straightline without changing speed—uniform motion. At firstthis seems contrary to our experience. Moving objectsalways seem to slow down and stop if nothing is doneto keep them moving. Is this not a violation of thisstatement of the law?

Our problem is that the objects with which we dealare not free. Friction acts on them and, if it is notopposed by other forces, “compels” these objects tochange from their state of uniform motion. We can testthe validity of the First Law, however, by considering themotion of objects in situations where friction is greatlyreduced. One can easily imagine, for example, that anice skater could glide on and on without ever slowingdown if friction could be eliminated totally (Fig. 3.1).

Figure 3.1. An ice skater could go on forever withouteffort if friction were not present.

Those who drive on icy roads are acutely aware of theconsequences of the law. It is a frightening experience toapproach an intersection, apply the brakes because of a redlight, and then proceed through without slowing down.Turning is also a problem because, without friction, the carcontinues in a straight line no matter how the wheels areturned. After two or three such experiences, one is easilyconvinced that the First Law of Motion is valid.

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Even so, the consequences of the law still catch usunaware. A common auto injury occurs when a passen-ger strikes the windshield when the car in which he is

riding suddenly stops. The passenger keeps moving inaccordance with the First Law of Motion (Figure 3.2).

A final example suggests another consequence ofthe law. A car makes a left turn at modest speed. Apackage next to the driver slides across the seat, awayfrom the center of the turn, and continues itsstraight-line motion in accordance with the First Law ofMotion, while the car turns under it (Fig. 3.3).

Incidentally, none of these examples “proves” thatthe First Law of Motion is valid, but all suggest that itmight be. Considering other consequences of the lawseems reasonable. As we gain additional experience wegain increasing confidence in the validity of the law.

Acceleration

Uniform motion in a straight line without changingspeed is the “natural” motion of free objects. Any objectthat is not in uniform motion is said to be accelerating.An object accelerates if its speed changes, either toincrease or decrease, or if its direction changes. It issometimes useful to assign specific words to describesome simple types of acceleration. Deceleration, forexample, denotes a decrease in speed whereas a direc-tion change is properly designated as a centripetal (cen-ter-seeking) acceleration. Any change from uniformmotion, however, is an acceleration (Fig. 3.4 and 3.5).

Figure 3.3. Why does the passenger feel “thrown” tothe outside of a turn?

Figure 3.4. Successive pictures, taken at equal time intervals, of a car in four different kinds of motion. Why do wesay that the car is accelerating in c and d but not in a and b?

Figure 3.2. Both drivers lose their heads in a rear-endcollision. Why?

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Figure 3.5. A coin rests on a moving turntable. Howdo you know it is accelerating?

Acceleration may be defined more precisely as therate at which speed or direction is changing. The accel-eration of a car might be designated properly as 5mi/hr/sec. This car would increase its speed from 30 to50 mi/hr in 4 sec. It would have the same amount ofacceleration if its speed increased from 10 to 30 mi/hr inthe same amount of time. If it slowed from 50 to 40

mi/hr in 2 sec the acceleration would be 25 mi/hr/sec;the negative sign denotes deceleration.

Force

Your intuitive understanding of force is probablyadequate for our present purposes. Force is simply apush or pull exerted on one object by another. A moresophisticated definition of force is implied by the FirstLaw of Motion: force is anything that causes accelera-tion. All accelerations are caused by forces. Forces areacting whenever an object moves faster, moves slower,changes direction, or experiences any combination ofspeed and direction change.

The kind of acceleration caused by a particularforce depends on the direction of the force. If a forcepushes on an object in the same direction as its motion,the object speeds up. The object slows down if the forceopposes its motion (Fig. 3.6). Lateral forces causechange in direction with the object turning toward thedirection of the force (Fig. 3.7).

The strength of forces is measured in pounds (lb)in the English system of units and newtons (N) in themetric system. The amount of acceleration produced bya particular force is determined partly by its strength.Stronger forces produce greater accelerations. If a par-ticular force causes an object to accelerate from 20 to 30mi/hr in 10 sec, a force twice as strong would cause thesame change in half the time. A force half as strongwould take 20 sec to produce the same effect.

Most objects we deal with are influenced by morethan one force. These forces may oppose each other sothat the resulting acceleration is reduced, or they mayact in the same direction so that the acceleration isgreater than for either one by itself. The sum of all theforces acting on an object is called the net force orresultant force. The strength and direction of the netforce determine the acceleration of the object.

Forces cannot be summed like ordinary numbers,however. Forces have both a magnitude (strength) and adirection. Such quantities, called vectors, can be repre-sented by an arrow whose length has been scaled to rep-resent the magnitude and whose direction is that of thepointed arrow. Two vectors can be added by forming aparallelogram with the two properly scaled and orientedvectors forming the adjacent sides. The diagonal of theparallelogram is the resultant force (sum of the two) andwill have both the proper length and direction.

Mass

Not all objects experience the same accelerationwhen acted upon by similar forces. An ordinary car, forexample, might be able to provide significant accelerationto a small empty trailer but considerably less when thetrailer is full. If a truck is loaded, the time and distancerequired for a safe stop increases significantly (Fig. 3.9).

The property of objects that determines how muchthey accelerate in response to applied forces is calledmass. If mass is large, acceleration will be less than ifmass were smaller. The smaller the mass, the greater theacceleration.Figure 3.6. Both pitcher and catcher exert forces that accel-

erate the baseball. In which direction is each force applied?

Figure 3.7. The puck slides in a circle on an air-hock-ey table without friction. In what direction is the forceexerted on the puck by the string? (Hint: Have youever seen a string that could push on anything?)

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Figure 3.8. Two vectors, A and B, are added to give theresultant, C. What is the resultant of D and E?

Mass does not depend on location. A particularforce causes the same amount of acceleration no matterwhere the object is located: near the earth, in interstel-lar space, or anyplace else (Fig. 3.10). If the sameobject experiences different accelerations at differentplaces, it is because the forces acting on it are different,not because its mass has changed.

To be useful, the concept of mass must be madequantitative. We want to know, for example, whether asack of potatoes has a mass of one kilogram or two kilo-grams. Quantities of mass are defined by comparison tosome arbitrarily defined standard. The standard kilo-gram is decreed to be the mass of a piece of platinum-iridium which is kept under the watchful care of theBureau Internationals des Poids et Measures at Sevresnear Paris. If you want to know whether you have onekilogram of potatoes, you must directly or indirectlycompare the mass of your potatoes with the mass of thispiece of metal.

To make this process practical, copies of the stan-dard kilogram are supplied to the bureaus of standards ofthe various nations. They, in turn, make copies—-someof which are split in halves, quarters, etc. You may haveseen a box of “weights” in a chemistry laboratory whichis the result of this process. One way to compare yourpotatoes to the standard mass is to place potatoes andstandard mass on opposite sides of a balance (scales) andlet gravity serve as a standard force. Put your potatoeson one side and keep adding standard masses to the otheruntil balance is achieved. Now add up the standardmasses you have used and this is equal to the mass of thepotatoes. You have made your comparison (indirectly)with the standard kilogram near Paris.

Length and time must also be given quantitativemeaning by comparison to standards. For many years

SPACE

EARTH

MOON

Figure 3.9. The same force applied to different objects produces different accelerations. Which of these trucks isempty? Which has the greater mass?

Figure 3.10. A rocket (or any other object) is just as hard to accelerate no matter where it is. Why?

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the standard meter was a long bar of metal kept along-side the standard kilogram. The second was defined assome fraction of the day. Today we have more precisestandards of time and distance which are based on cer-tain characteristics of atoms. The equations we presentin this book are usually presented in a form whichrequires that a certain consistent set of units be adoptedwhen using the equation. The metric system usesmeters for length, seconds for time, and kilograms formass; the English system uses feet for length, secondsfor time, and slugs for mass. (While almost all civilizednations have adopted the metric system, the UnitedStates remains an official user of the foot-pound-secondsystem. This will almost surely change because itworks to our disadvantage in world trade.)

The Second Law of Motion

Perhaps briefly summarizing what we know aboutmotion so far will help:

1. If an object is left to itself, it will remain at restor move with its initial uniform motion.

2. Forces cause objects to accelerate. The strongerthe net force, the greater the acceleration.

3. Accelerations are less if mass is larger.4. Acceleration is in the same direction as the net

applied force (forward, backward, sideways, orsome combination of these).

The second, third, and fourth of these statementsconstitute the Second Law of Motion. In addition, thelaw specifies the exact relationship between the mass ofan object, the strength of the net force applied to it, andthe amount of acceleration caused by the force. Therelationship is

acceleration 5 net forcemass

or, equivalently,

net force 5 mass 33 acceleration.

With the First and Second Laws of Motion, youcan begin to study the motion of anything you observe.Remember, the important question is not why an objectkeeps moving but why its motion changes. This ques-tion directs our attention to a search for forces and theircauses. An understanding of the forces enables us todetermine if the accelerations we observe are consistentwith the forces and with the object’s mass. If they are,we can go on to other interesting problems; if not, wehave more to learn.

The Third Law of Motion

Forces act on all objects. To understand objects’motion, or lack of motion, we must consider whereforces come from, in what situations they occur, andwhat determines their strength and direction.Otherwise, we can neither explain nor predict motion.

The first important observation about forces is thatthey occur only when two things interact with eachother. Nothing can exert a force on itself. For example,the wheels of a car touch the road. If the interactionsbetween drive wheels and road do not occur, perhapsbecause of ice on the road, there is no force and the cardoes not accelerate. A boat propeller touches the water;an airplane propeller, the air. The forces that acceleratea rocket result from the contact between the rocket itselfand the fuel that burns inside. No object or system thatcan exert a net force on itself has ever been found orinvented. Forces occur only when two objects are asso-ciated with each other, the most common associationbeing actual contact.

The next important observation is that two forcesact in every interaction, one on each of the interactingobjects. In some cases, the two forces are both appar-ent. As a man steps from a small rowboat to a dock, heis accelerated toward the dock and the boat acceleratesin the opposite direction. A rifle recoils (accelerates)whenever a bullet is fired.

Sometimes, however, the second force is less obvi-ous and we may not recognize its presence. When youstart to walk, for example, the force that accelerates youcomes from the interaction between your foot and thefloor. You push backward on the floor (using your legmuscles), the floor pushes forward on you, and youaccelerate in the direction of this force exerted on yourfoot by the floor.

The forward force on your foot in this example isobvious. It causes your foot and its attachments toaccelerate. The backward force is not quite so apparent.Nothing seems to accelerate in that direction. We usu-ally do not notice that the floor is rigidly attached to theearth, and so its effective mass is quite large. The flooris, in fact, accelerated backward, but the amount ofacceleration is immeasurably small because of thefloor’s large mass. The presence of this backward forcewould easily be revealed if the floor were covered withmarbles. Their backward acceleration as you walked onthem would make the backward force readily apparent.

By now you should have noticed that the two forcesthat interacting objects exert on each other always act inopposite directions. When a man steps from a rowboat, heis accelerated one way while the boat moves in the oppo-site direction. A bullet is fired in a particular direction,and the gun recoils oppositely. You push backward on thefloor and the floor pushes forward on you. The two forcesin every interaction are always oppositely directed.

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It can be shown by careful measurements that thetwo forces in any interaction have the same strength—arule always obeyed by nature. For example, if youforcefully kick a stone, your toe receives the benefit ofa force that has the same bone-breaking strength. If youkick more gently, the force on your toe is also more gen-tle by exactly the same amount (Fig. 3.11).

You can probably imagine how this rule might betested. Arrange for two objects, whose masses you knowfrom another experiment, to interact with each other;measure the accelerations caused by the forces of inter-action; and use the Second Law of Motion to calculatethe forces. Thousands, perhaps millions, of experimentsof this kind have been performed since Newton first sug-gested the rule. In every case, the forces the interactingobjects exert on each other have been shown to haveexactly the same strength (Fig. 3.12).

The properties of forces described above are col-lectively known as the Third Law of Motion, which isstated as follows:

All forces result from interactions between pairsof objects, each object exerting a force on theother. The two resulting forces have the samestrength and act in exactly opposite directions.

As you can see, the Third Law of Motion is a ruleabout forces. It is a law of “motion” only to the extentthat forces and motion are related through the SecondLaw of Motion. Nevertheless, the law seems to beobeyed by all the forces in nature that can be studied indetail. There are apparently no exceptions.

Notice that the Third Law does not tell everythingabout forces. It gives no information about how strongthe forces will be for any given interaction. This infor-mation is expressed by force laws that describe thekinds of interactions that occur in nature and the result-ing forces. These are described in the next chapter.

Applications

The First Law of Motion can be used to“explain” auto whiplash and windshield injury, thesensation of being thrown outward during a turn, thealmost effortless motion of an ice skater, and othercommon experiences.

The Second Law of Motion can be used to“explain” the nearly circular motion of the planets, whysliding objects slow down, why it is hard to stop or turnon slick roads, the behavior of electrons in a TV tube,the operation of electric and gasoline motors, why it iseasier to accelerate a motorcycle than a truck, andmuch, much more. Indeed, every mechanical deviceinvolving internal or external motion is based on theSecond Law.

The Third Law of Motion can be used to “explain”the operation of a rocket engine, the “kick” of a rifle orshotgun, the operation of a jet or propeller-driven air-craft, the motion of a boat when a person steps off it,and many other phenomena.

In each case, the Third Law of Motion describessome features of the relevant forces. The resultingmotions are then predicted by the Second Law of Motion.

Figure 3.11. Identify the interaction, the two resultingforces, and the accelerations that are produced when aperson kicks a rock.

Figure 3.12. Why is gravel thrown backward when a car accelerates? What force accelerates the car?

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Summary

The three laws of motion provide important insightsas we attempt to understand the changes that occuraround us. The First Law directs our attention tochanges from uniform motion, or accelerations. TheSecond Law suggests that these are caused by forces andthat the motion change in any situation is determined bytwo factors: the strength and direction of the appliedforces and the mass of the accelerating object. Finally,the Third Law identifies interactions as the source of allforces. It also provides certain details about the forcesresulting from any interaction that we might discover.

You may have noticed that all three laws are need-ed to understand the motion of any real object. In everycase we must direct our thoughts to acceleration andforce and to the relationship between them. Forces inturn lead us directly to interactions. Each law is of lim-ited use without the insight that the others provide.

The genius and insight of Isaac Newton wererequired to discover all three laws at once. Newton ledthe way for each of us to discover the order and harmo-ny in the motions we observe.

STUDY GUIDEChapter 3: Laws Governing Motion

A. FUNDAMENTAL PRINCIPLES1. The First Law of Motion: Every object continues

in its state of rest, or of uniform motion in a straightline with unchanging speed, unless compelled to dootherwise by forces acting upon it.

2. The Second Law of Motion: (a) Forces causeobjects to accelerate. The stronger the net force,the greater the acceleration. (b) Accelerations areless if mass is larger. (c) Acceleration is in thesame direction as the net applied force (forward,backward, sideways, or some combination ofthese). More precisely, acceleration = force/mass.

3. The Third Law of Motion: All forces result frominteractions between pairs of objects, each objectexerting a force on the other. The two resultingforces have the same strength and act in exactlyopposite directions.

B. MODELS, IDEAS, QUESTIONS OR APPLICA-TIONS

1. The Newtonian Model (sometimes, theNewtonian Synthesis): The model based onNewton’s three laws of motion and the UniversalLaw of Gravitation which explains the motions ofthe heavens as well as the terrestrial motions ofcommon experience. The Newtonian Model whenapplied to the motions of the planets replaces themedieval model which placed the earth at the cen-ter of the solar system and the universe.

C. GLOSSARY1. Acceleration: Change from uniform motion.

Acceleration in this broader sense may be acceler-ation (speeding up), deceleration (slowing down),centripetal (change of direction) or a combinationof these.

2. Force: A push or pull. All forces result from inter-actions between two objects (Third Law), so thatthe terms “force” and “interaction” are often usedinterchangeably.

3. Friction: A force arising from contact betweensurfaces of materials that prevents or retards rela-tive motion of the surfaces.

4. Mass: A characteristic of objects which determinesthe degree to which they can be accelerated byapplied forces. Mass is also a characteristic ofobjects that determines the strength of their gravi-tational interaction with matter, specifically withother objects with mass.

5. Net Force (sometimes, resultant force): The sin-gle force which can be used to replace two or moreindividual forces on an object for purposes of deter-mining its motion.

6. Uniform Motion: Motion in a straight line withunchanging speed.

7. Standard: An arbitrarily chosen object of compar-ison for purposes of defining units of measurement.Units of length, mass, and time are defined by com-parison to standards. For example, the standardkilogram is a particular piece of platinum-iridiummetal belonging to the International Bureau ofWeights and Measures near Paris.

D. FOCUS QUESTIONS1. In each of the following situations:

a. Describe what would be observed.b. Name and state in your own words the funda-mental principle(s) that could explain what wouldhappen.c. Explain what would happen in terms of the fun-damental principle(s).

(1) A styrofoam ball and a steel ball of equalsize are each suspended from a cord and sub-jected to about the same amount of force froma blunt-nosed dart fired horizontally from aspring-loaded gun.

(2) Suppose an elephant and an ant are bothmoving at the same speed on a level, friction-less surface: which would stop first (or nei-ther)? Assume air friction to be unimportantfor both.(3) A large truck moving at high speed col-lides with an unsuspecting mosquito thatbefore the collision is hardly moving at all.

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What happens to the motion of both the mos-quito and the truck?

(4) Suppose a man jumps forward toward adock from a small boat that is not securelyanchored. What happens to the horizontalmotion of both the man and the boat if the boathas less mass than the man?

E. EXERCISES3.1. Suppose you have a friend who does not

believe the First Law of Motion is true. How would youproceed to convince the friend otherwise?

3.2. If an elephant and an ant are both moving atthe same speed on a level, frictionless surface, whichwould stop first? Assume air friction to be unimportantfor both.

3.3. An unrestrained child is standing on the frontseat of a car traveling at 20 mi/hr in a residential neigh-borhood. A dog runs in front of the car and the driverquickly and forcefully applies the brakes. The child’shead strikes the windshield.

(a) Explain this result in terms of the First Law ofMotion.(b) If the car stopped before the child reached thewindshield, with what speed would the child’s headstrike the windshield?

3.4. Using the First Law of Motion, explain why apassenger in a turning car feels “thrown” away from thecenter of the turn.

3.5. State the First Law of Motion in your ownwords. Explain its meaning.

3.6. What do the words “uniform motion” mean aspart of the First Law of Motion?

3.7. A car travels in a large circle (in a parking lot,for example) without changing speed. Is the car in uni-form motion? Explain your answer.

3.8. In each of the following situations, describe(1) what actually happens or would be expected to hap-pen and (2) how these results can be accounted for bythe First Law of Motion.

(a) A car is struck from behind by a faster movingvehicle. A passenger later complains of whiplashinjury.(b) A car experiences a head-on collision with alamppost. A front-seat passenger is not wearing aseat belt.(c) A ball is placed on a level table fixed to the floorof a train at rest in a station. The train suddenly

starts moving. (d) A ball is placed on a level table fixed to the floorof a train which is moving with uniform motion.The train suddenly stops.(e) Same as (d) except the train speeds up.(f) Same as (d) except the train goes around a curvein the track.

3.9. The driver of a car has three “accelerators” (con-trols that can cause the car to accelerate). What are they?

3.10. Suppose you see an object traveling in a cir-cle with constant speed. What can you say for sureabout the force or forces acting upon it?

3.11. Describe an experiment you could performthat would determine which of two objects has the larg-er mass. Be sure that your experiment is consistent withthe definition of mass given in this chapter.

3.12. A constant force is continuously applied to anobject that is initially at rest but free to move withoutfriction. No other forces act on the object. Describewhat would be observed under these conditions andexplain how the observed results can be accounted forby the Second Law of Motion. Finally, explain why realobjects—-cars for example—-do not behave in thisway.

3.13. Describe the three simple types of accelera-tion which are governed by the Second Law of Motion.

3.14. Does a car accelerate when it goes up a hillwithout changing speed? Explain your answer.

3.15. Describe the accelerations which occur as anelevator rises, starting from rest at the first floor andstopping at the twentieth floor.

3.16. Imagine an object resting on a horizontal sur-face where there is no friction (an air-hockey table, forexample). A force is applied to it so that it accelerates,sliding along the surface. Now imagine that the wholeapparatus is taken to the moon where the same experi-ment is performed using the same object and the sameforce. How would the acceleration of the object nearthe moon compare with that near the earth?

3.17. Now suppose that the object in the previousquestion is taken to a place, a long way from the earthor moon, where it is weightless. Again, the same forceis applied to it (using a small rocket engine, for exam-ple). What does it do?

3.18. State the Second Law of Motion and explainits meaning.

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3.19. An object is traveling on a smooth horizon-tal surface where the friction can be ignored. A force isapplied (perhaps by a rocket engine or propellerattached to the object) in a direction opposite to theobject’s velocity. Describe what would happen andexplain how this is consistent with the Second Law ofMotion.

3.20. A rocket in deep space requires less and lessforce to accelerate it at the same rate, even though thegravitational and frictional forces on it do not change.What can you conclude?

3.21. A car turns a corner at constant speed. Isthere a force acting on the car? Explain your answer.

3.22. Show how the recoil or “kick” of a rifle orshotgun can be accounted for by the Second and ThirdLaws of Motion.

3.23. How would the accelerations of a gun andbullet compare if the gun had 500 times more mass thanthe bullet? How would they compare if the gun and bul-let had the same mass? Explain your answers in termsof the Second and Third Laws of Motion.

3.24. Describe the force which causes a car toaccelerate as it starts from rest. Identify the importantinteraction, describe the two forces in the interaction,and indicate the directions of both forces. Finally,describe the resulting accelerations.

3.25. Explain the operation of a rocket engine interms of the Second and Third Laws of Motion.

3.26. (a) Describe what happens when a man jumpsfrom a small boat if the boat is not securelyanchored.(b) Explain how the observed result can beaccounted for by the Second and Third Lawsof Motion.(c) What would be different if he jumped froma large boat? Why would this situation be different?

3.27. A balloon is filled with air and then released.(a) What do you imagine the balloon does?(b) Explain the imagined motion of the bal-loon by using the Second and Third Laws ofMotion.

3.28. A truck moving at a high speed collides witha mosquito.

(a) Describe and compare the forces in theinteraction.

(b) If the truck hits the mosquito from theblind side, so that the mosquito couldn’t getready, could it exert a greater force on the mosquito than the mosquito exerts on thetruck? Explain your answer in terms of a fundamental law.

3.29. Describe the force (or forces) which causeyou to accelerate when you start to walk. That is, iden-tify the interaction, describe the two forces in the inter-action, and indicate the directions of both forces.Finally, describe the resulting accelerations.

3.30. Do we arrive at the Third Law of Motionthrough an inductive or a deductive process? Can thelaw be proved to be true? How could it be proved to befalse?

3.31. Person X stands on a level, frictionless sur-face. Which is true?

(a) X cannot start moving, but if moving canstop.(b) X cannot change horizontal speed or direc-tion.(c) X can change speed or direction gradually.(d) X can change speed, but can’t stop.(e) X can change horizontal motion via verti-cal motion.

3.32. While riding your bicycle you collide head-onwith a moving car. The acceleration you experience is

(a) the same as that of the car(b) slightly greater than that of the car(c) slightly less than that of the car(d) much less than that of the car(e) much greater than that of the car

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