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WORKED EXAMPLES OF MATHEMATICAL PROBLEMS PHYSICS IN FOCUS PRELIMINARY COURSE

1Copyright McGraw-Hill Australia Pty Ltd

ModuleSyllabus statement #

3rd column dot-point # Equation(s)

Example numbers

8.2 The world communicates

1 6 v = f 1, 23 4 (2nd column)

Id

12 3, 4

4 4 ..calculate the refractive index of glass or Perspex

5

5 vv

ir

1

2

=

sinsin 6, 7

8.3 Electrical energy in the home

2 2E

F

q= 8, 9

4R

V

I= 10, 11

4 4 P = VI 12, 13

4 Energy = VIt 14, 15

8.4 Moving about 1 2v

r

tav=

16, 17

2 4 vector diagrams for resultant velocity, acceleration and force

18, 19, 20

6 F = ma 21, 227

Fmvr=

2

23, 24

3 1E mvk =

1

22

25, 26

1 W = Fs 26, 27, 28

4 1 p = mv 29, 30

1 Impulse = Ft 31, 32

Points to note:

1. It is important to show the equation being used before you substitute values.

2. Keep all your work neat (as you would in maths). If the answer is wrong but the mistake can be traced back to a minor mathematical error, some or most marks would still be awarded in most cases.

3. Equations link many things. If a question asks you to describe how A affects B, think of an equation with A and B in it.

4. These worked examples show the answers with the correct number of signi cant gures. This is the least number of signi cant gures in given or known values in the example. Round off to this number of signi cant gures only in the nal answer.

5. Ensure that the correct units are given with the answers.

From 3rd column dot points with solve and/or analyse

WORKED EXAMPLES OF MATHEMATICAL PROBLEMS



6. Units are not included within the calculation stages, but, again, ensure that the right unit is given at the end.

7. The nal answer to a Physics question usually corresponds to something real. If your answer looks impossible or seems silly, check it through carefully. For example, if you nd that the speed of an object is 5.0 109 m s1, it is likely that you got it wrong! (Why?)



Use the speed of light in a vacuum, c, as 3.0 108 m s1 and the speed of sound in air as 330 m s1 in the following questions.

1. Compare the range of wavelengths of sound that a human can hear (frequency range 20 Hz to 20 kHz) to the range of wavelengths that a human can see (frequency range 4.3 1014 Hz to 7.5 1014 Hz).

2. Tsunamis are reported to travel across the deep ocean at speeds of around 800 km hr1 appearing as slight rises in the water level rather than breaking waves. They often go unnoticed by ocean-going vessels. Find the wavelength of a tsunami in the deep ocean if its period T (time to complete one whole wave cycle) is 5 minutes.

3. The noise of an aircrafts jet engines from a few metres away can be deafening. As with light, the inverse square law also applies to sound. By how much does the intensity of the aircrafts noise change if an observer moves from a distance of 10 m to 200 m away?

4. In a science fi ction story, the Earth is moved further away from the Sun to prevent global warming. The Earth is moved from its present distance of 150 million km from the Sun to 200 million km. What is the fraction of the original intensity of sunlight once the Earth has been moved?

Examples

check solution

check solution

200 m

10 m check

solution

Earth moved

150 million km

Sun 200 million km

check solution



5. In an investigation to fi nd the refractive index of a Perspex block, measurements were made using a ray of light from a ray box. These measurements are shown in the diagram below.

Using these measurements, fi nd the refractive index of the Perspex.

6. A ray of light is slowed to half its original speed when it enters a second medium. The rays angle of incidence is 40. Find its angle of refraction, r.

7. For a ray of light entering a medium where its speed increases, show using equations how the value of the angle of incidence, i, can be used to fi nd the relative refractive index of the mediums when the angle of refraction, r, is 90 exactly.

8. What is the magnitude and direction of the electric fi eld, E, required to perfectly balance a single electron in space against its own weight? The mass of one electron is 9.1 1031 kg and its charge is 1.6 1019 C.

9. A mass of 5.0 106 kg is accelerated at 4.0 mm s2. What is the charge on this mass if the electric fi eld strength is 8.5 N C1? Ignore gravity in this question.

10. What is the current fl owing through a long-distance electricity transmission wire when the potential difference across a segment of wire is 2.25 102 V and the resistance of the segment is 1.50 ?

300

air 450

Perspex

check solution

400

v1

v2 = v1

r

12

check solution

check solution

check solution

check solution

check solution



11. (a) When 60.0 A of current fl ows through the wires connecting a cars battery to its starter motor, a potential difference of 1.5 V exists across the ends of the wire. What is the resistance of these wires?

(b) Suggest a reason why these wires are made as thick and as short as possible.

12. Compare the maximum power that can be provided by (a) a 240 V source, and (b) a 12 V source, when they are connected to appliances using a circuit breaker that can have a maximum current of 10.0 A fl owing through it.

13. In an investigation using a 12 V model heating coil it was noticed that the current through the coil was proportional to the voltage applied. What changes occur to the power output of the heating coil when the voltage is doubled?

14. A household uses 14 lights rated at 100 W each for an average of 5 hours per day. Electricity costs 15 cents per kilowatt hour. What is this households electricity bill for the use of these lights for 90 days?

15. During peak usage, the electrical power consumed by the state of New South Wales was estimated as being 12 000 MW. If the average cost to all electricity consumers during this peak period was 12 cents per kilowatt hour, how much money is being charged each hour by the electricity companies?

16. A plane fl ies from Alice Springs due north for 200 km and then turns east and fl ies for 300 km before fi nally turning west and fl ying for 500 km. It then lands. The entire journey takes 2 hours.

(a) Find the planes displacement from Alice Springs. (b) Compare qualitatively and quantitatively the planes average speed with its

average velocity.

17. Give an example of a sporting event in which the average velocity of a competitor is zero (or nearly zero) but their average speed is relatively high. Write a question using your example to submit to the class for discussion.

18. While analysing their motion over a map of the world, the crew of a ship found that they were moving to the north at 17 km hr1 and to the west at 5.0 km hr1 simultaneously. (The northerly and westerly motions are known as components of the motion.) What is the actual velocity of the ship?

19. An astronaut is hovering above the Moons surface in a new hoversuit. The astronaut has several forces being exerted on her as shown in the diagram.

Find the astronauts acceleration by fi rst fi nding the net force acting on her.

check solution

check solution

check solution

check solution

check solution

check solution

check solution

check solution

pulling on rope 25 N mass of astronaut (and hoversuit)

= 100 kg rope

weight thrust from hoversuit 160 N 200 N check

solution



20. Three spring balances were arranged on a table attached to a moveable block as shown. The block is not accelerating. Two of the springs are pulling with forces shown. The third springs scale cannot be read. Calculate the force in this spring.

21. A toy train consists of three carriages being pulled by one engine, as shown below. The engine has a mass of 2.0 kg and each carriage has a mass of 1.5 kg. The train is accelerating at 0.20 m s2. Find:

(a) the net force on the train overall; and (b) the net force acting on the carriage marked X.

22. A 5.0 104 kg truck is accelerating along a level road at 1.5 m s2. The force of friction against its motion totals 9.0 kN. What is the useful force being exerted on the truck by its engine?

23. A car with a mass of 1.8 103 kg travelling at 25 m s1 enters a curve with a radius of 120 m.

(a) What is the centripetal force required to keep the car moving on the road at that speed?

(b) Describe the cars motion if a section of the curved road was covered by frictionless ice.

(c) Describe how the centripetal force required is related to the speed of a car moving around a bend and discuss how this relationship is the physics behind so many car crashes.

24. Find the maximum speed which a 1.5 tonne car can negotiate a curved section of road that has a radius of curvature of 200 m if the tyres can exert a maximum sideways force on the road of 3.0 kN before beginning to slide?

25. Find the kinetic energy of a 250 kg motorbike moving at 30.0 m s1.

?? N

10.0 N 10.0 N 450 45

0

a = 0.20 m s2

2.0 kg 1.5 kg 1.5 kg 1.5 kg

X

check solution

check solution

check solution

check solution

check solutioncheck

solution



26. A net force of 800 N acts on the motorbike in Example 25 for a distance of 100 m. (a) Find the work done by this net force: (b) hence fi nd the motorbikes new speed.

27. The brakes of a car or truck transform kinetic energy into heat. Assume that this transformation is 100% complete.

(a) Find the amount of heat energy produced by the brakes when a 40 tonne truck is stopped from a speed of 25 m s1; and

(b) Find the distance over which a force of 2.0 104 N must act in order to do the same amount of work as the brakes in part (a).

28. If the truck in Example 27 is stopped in a distance of 3.0 m what average force must be exerted?

29. (a) Find the momentum of the truck in Example 27. (b) Find the impulse on the truck when it is stopped. (c) Using equations for momentum and impulse, fi nd the time taken to stop the

truck using your answer from Example 28.

30. On the same axes, sketch a graph showing the relationship between: (a) an objects momentum as its speed varies; and (b) an objects kinetic energy as its speed varies.

31. A rocket motor exerts a force of 45 kN for 20 s. What is the impulse given by the motor?

32. Explain how a future spaceship motor that exerts a small force over a long time could produce the same increase in speed as a motor exerting a large force over a short time.

check solution

check solution

check solution

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check solution

check solution



Use the speed of light in a vacuum, c, as 3.0 108 m s1 and the speed of sound in air as 330 m s1 in the following questions.

Example 1

Compare the range of wavelengths of sound that a human can hear (frequency range 20 Hz to 20 kHz) to the range of wavelengths that a human can see (frequency range 4.3 1014 Hz to 7.5 1014 Hz).

SolutionFirst, fi nd the wavelengths of sound for sound frequencies of 20 Hz and 20 kHz:

vsound

= 330 m s1

v f

v

f

f

f

=

=

=

=

=

=

for Hz

330

20m

for kHz

20

16 5

20

.

=

=

330

20 10

m or 1.7 cm

3

1 7 10 2.

Next, fi nd the wavelengths for light:

vlight

= 3.0 108 m s1

v f

v

f

f

=

=

=

=

=

for Hz

3.0 10

4.3 10

8

14

4 3 1014.

77.0 10 m

for Hz

3.0 10

1

7

8

=

=

f 7 5 10

7 5

14.

.

00

m

14

= 4 0 10 7.

The highest pitch a human can hear has a frequency 1000 times greater than the frequency of the lowest audible pitch. The highest frequency light (violet) a human can see has only twice the frequency of red light, which has the lowest visible frequency. The wavelengths of sound waves are around 100 000 times longer than the wavelengths of light.

Solutions



Example 2

Tsunamis are reported to travel across the deep ocean at speeds of around 800 km hr1 appearing as slight rises in the water level rather than breaking waves. They often go unnoticed by ocean-going vessels. Find the wavelength of a tsunami in the deep ocean if its period T (time to complete one whole wave cycle) is 5 minutes.

SolutionThe frequency f is found using f

T=

1

fT

T

=

=

=

=

=

1

given 5 minutes

5 60 s

300 s

f1

300Hzz

and

1

300

v f

v

f

=

=

=

( ) ( )

=

800 1000 60 60

6 67. 1104 m or 66.7 km

The long wavelength is another reason why tsunamis often go unnoticed in the open ocean.



Example 3

The noise of an aircrafts jet engines from a few metres away can be deafening. As with light, the inverse square law also applies to sound. By how much does the intensity of the aircrafts noise change if an observer moves from a distance of 10 m to 200 m away?

SolutionThe intensity I is given by the inverse square law. The ratio of the intensities is also given by the same law, so that:

Id

d d

I

I

d

d

= =

=

( )( )

=

1200

2 1 2

2

1

1

2

2

2

and 10 m and m

110

2001

40

2

=

The intensity is 1/40th of the original.

200 m

10 m



Example 4

In a science fi ction story, the Earth is moved further away from the Sun to prevent global warming. The Earth is moved from its present distance of 150 million km from the Sun to 200 million km. What is the fraction of the original intensity of sunlight once the Earth has been moved?

SolutionA similar approach to that used in Example 3 can be taken in this example:

d1 = 150; d

2 = 200; let I

1 = 1

Id

d

I

I

d

d

=

=

( )( )

12 1

2

1

1

2

2

and 5 million km and d = 200 million km

22

2150

2000 56

=

= . i.e. the intensity of sunlight is 56% of the original

2

Earth moved

150 million km

Sun 200 million km



Example 5In an investigation to fi nd the refractive index of a Perspex block, measurements were made using a ray of light from a ray box. These measurements are shown in the diagram below.

Using these measurements, fi nd the refractive index of the Perspex.

SolutionUsing Snells law and being careful to read the angles from the normal, NOT as shown in the diagram:

Using n2 = refractive index of Perspex and n

1 = 1.0 (refractive index of air):

sin

sin

sin

sinsin

sin.

i

r

n

n

nn i

r

=

=

=

=

2

1

21

60

451 2

300

Air 450

Perspex

600

300

Air 450

450

Perspex



Example 6

A ray of light is slowed to half its original speed when it enters a second medium. The rays angle of incidence is 40. Find its angle of refraction, r.

SolutionUsing:

v

v

i

r1

2

=

sin

sin, where v

v1

2

2=

sin

sin

sinsin

.

sin .

i

r

r

r

=

=

=

=

=

2

40

20 321

0 321

19

1

400

v1

r

v2 =

1

2v

1



Example 7

For a ray of light entering a medium where its speed increases, show using equations how the value of the angle of incidence, i, can be used to fi nd the relative refractive index of the mediums when the angle of refraction, r, is 90 exactly.

SolutionWhen r = 90, sin r = 1.00

Also, when this occurs, angle i = c, the critical angle

sin

sin

i

r

n

n=

2

1

Using

sin

.,

c n

n=

2

11 00i.e. the relativve refractive indices



Example 8

What is the magnitude and direction of the electric fi eld, E, required to perfectly balance a single electron in space against its own weight? The mass of one electron is 9.1 1031 kg and its charge is 1.6 1019 C.

SolutionBalance F = Eq (from E

F

q=

) with W = mg (the weight of the electron):

Eq mg

Emg

q

=

=

=

=

9 1 10 9 8

1 6 10

5 6 10

31

19

11

. .

.

. N C down (a very small E field)1



Example 9A mass of 5.0 106 kg is accelerated at 4.0 mm s2. What is the charge on this mass if the electric fi eld strength is 8.5 N C1? Ignore gravity in this question.

SolutionBalance F = ma with F = Eq:

ma Eq

q

=

= 5 0 10 4 0 10 8 56 3. . . kg m s N C2 1

qq =

=

5.0 10 kg m s

N C

6 2

-1

4 0 10

8 5

2 4 1

3.

.

. 0 9 C



Example 10

What is the current fl owing through a long-distance electricity transmission wire when the potential difference across a segment of wire is 2.25 102 V and the resistance of the segment is 1.50 ?SolutionUse Ohms law: V = IR

IV

R=

=

=

2 25 10

1 50150

2.

.A



Example 11

(a) When 60.0 A of current fl ows through the wires connecting a cars battery to its starter motor, a potential difference of 1.5 V exists across the ends of the wire. What is the resistance of these wires?

(b) Suggest a reason why these wires are made as thick and as short as possible.

Solution(a) Use Ohms law: V = IR

RV

I=

=

=

1 5

60 02 5 10 2

.

..

(b) The shorter and thicker the wires, the lower the resistance of the wires. A low resistance will minimise the voltage drop when a large current fl ows, so that as much of the batterys voltage as possible is supplied to the starter motor.



Example 12

Compare the maximum power that can be provided by (a) a 240 V source, and (b) a 12 V source, when they are connected to appliances using a circuit breaker that can have a maximum current of 10.0 A fl owing through it.

SolutionUsing P = VI, it can be seen that a 240 V source can deliver 20 times more power than a 12 V source for the same current fl ow. This is why many trucks use two 12 V batteries connected in series to give 24 V for their electrical systems, as the starter motor requires too much power for a 12 V system.

This topic is covered further in the HSC course.



Example 13

In an investigation using a 12 V model heating coil it was noticed that the current through the coil was proportional to the voltage applied. What changes occur to the power output of the heating coil when the voltage is doubled?

SolutionAssuming that the heating coil is an ohmic resistance, doubling the voltage will also double the current fl owing through it.

Using P = VI, doubling both V and I results in a four-fold increase in the power.



Example 14

A household uses 14 lights rated at 100 W each for an average of 5 hours per day. Electricity costs 15 cents per kilowatt hour. What is this households electricity bill for the use of these lights for 90 days?

Solution

Electrical energy consumed:

E VIt

power time

=

=

= ( ) =

14 100 5 90W hr/day days

6630 000

630

W h

kW h=

cost at 15c per kWh: cosst =

=

630 0 15

94 50

$ .

$ .



Example 15

During peak usage, the electrical power consumed by the state of New South Wales was estimated as being 12 000 MW. If the average cost to all electricity consumers during this peak period was 12 cents per kilowatt hour, how much money is being charged each hour by the electricity companies?

Solution

Energy = power time = 12 000 106 watts 1 hour = 12 000 106 watt-hours = 12 000 103 kW h = 1.20 107 kW h Cost = energy consumed in kW h cost per kW h = 1.20 107 $0.12 = $1.4 million dollars



Example 16

A plane fl ies from Alice Springs due north for 200 km and then turns east and fl ies for 300 km before fi nally turning west and fl ying for 500 km. It then lands. The entire journey takes 2 hours.(a) Find the planes displacement from Alice Springs.(b) Compare qualitatively and quantitatively the planes average speed with its

average velocity.

Solution(a) Vector addition gives:

(b) The planes average speed is distance travelled (200 km + 300 km + 500 km) = 1000 km divided by the time taken (2 hours).

The planes average speed is 500 km hr1. The planes average velocity is its displacement (283 km NW) divided by the time

taken (2 hours). The planes average velocity is 141.5 km hr1 NW. The magnitude of the average velocity is smaller than the average speed.

500 km

Plane lands here

200 km 300 km 200 km

Alice Springs

1

2

3

200 200

80000

283

2 2+

=

= km NW



Example 17

Give an example of a sporting event in which the average velocity of a competitor is zero (or nearly zero) but their average speed is relatively high. Write a question using your example to submit to the class for discussion.

SolutionSporting events could include: cycle racing; marathons; 400 m races; car races or any other event where the start and fi nish are close together.



Example 18

While analysing their motion over a map of the world, the crew of a ship found that they were moving to the north at 17 km hr1 and to the west at 5.0 km hr1 simultaneously. (The northerly and westerly motions are known as components of the motion.) What is the actual velocity of the ship?

SolutionVector addition of the two components of motion is required. The resultant is the actual velocity of the ship.

5.0 km hr1 17 km hr1

17 5 0

314

17 7

2 2

.

. km hr1

The direction is giveen by finding the angle Q

Q

Q

tan.

tan

5 0

17

11 5 0

1716

.

o (to the nearest degree)

So the shipss velocity is 18 km hr N16 W1 o



Example 19

An astronaut is hovering above the Moons surface in a new hoversuit. The astronaut has several forces being exerted on her as shown in the diagram.

Find the astronauts acceleration by fi rst fi nding the net force acting on her.

SolutionA vector diagram is used to fi nd the net force, F:

pulling on rope 25 N mass of astronaut (and hoversuit)

= 100kg rope

weight thrust from hoversuit 160 N 200 N

weight thrust from hoversuit 100 N 200 N

rope 25 N

1

2

3

40 25

2225

47 2

25

2 2+

=

=

=

. N

with an angle:

tan1440

32= o

The acceleration of the astronaut is:

aF

m=

=

=

47.2 N

100 kg

m s up and 32 to t2 o0 47. hhe left



Example 20Three spring balances were arranged on a table attached to a moveable block as shown. The block is not accelerating. Two of the springs are pulling with forces shown. The third springs scale cannot be read. Calculate the force in this spring.

SolutionThe two spring balances with 10 N force in them are placed symmetrically, both contributing a force down the page of 10cos45 = 7.07 N

The total force down the page is therefore 2 7.07 N = 14.1 N.(The sideways components of the forces in these two springs cancel each other out.)

As F = 0, the force in the third spring is 14.1 N

?? N

10.0 N 10.0 N 450 45

0



Example 21

A toy train consists of three carriages being pulled by one engine, as shown below. The engine has a mass of 2.0 kg and each carriage has a mass of 1.5 kg. The train is accelerating at 0.20 m s2. Find:(a) the net force on the train overall; and(b) the net force acting on the carriage marked X.

Solution

(a) Use F = ma = (2.0 + 1.5 + 1.5 + 1.5 kg) 0.20 m s2

= 1.3 N(b) Carriage X has a mass of 1.5 kg.

Again use: F = ma = 1.5 kg 0.20 m s2

= 0.30 N

a = 0.20 m s2

2.0 kg 1.5 kg 1.5 kg 1.5 kg

X



Example 22

A 5.0 104 kg truck is accelerating along a level road at 1.5 m s2. The force of friction against its motion totals 9.0 kN. What is the useful force being exerted on the truck by its engine?

Solution

Ftruck

= mtruck

atruck

= 5.0 104 kg 1.5 m s2

= 7.5 104 N

As F = Fengine

Ffriction

(as friction is in the opposite direction to the force from the engine)

Fengine

= F + Ffriction

= 7.5 104 + 9.0 103 N

= 8.4 104 N



Example 23

A car with a mass of 1.8 103 kg travelling at 25 m s1 enters a curve with a radius of 120 m. (a) What is the centripetal force required to keep the car moving on the road at that

speed?(b) Describe the cars motion if a section of the curved road was covered by

frictionless ice.(c) Describe how the centripetal force required is related to the speed of a car

moving around a bend and discuss how this relationship is the physics behind so many car crashes.

Solution

(a) Fmv

rm r m v

=

= = =

2

31 8 10 120 25where: kg; m. ; ss

F

N towards the

1:

.

.

=

=

1 8 10 25

1209 4 10

3 2

3 centre of the arc made by the curve

(b) Frictionless ice (known as black ice to drivers) results in the car continuing in a straight line rather than a curve. Cars often run off the road when they hit ice even when going very slowly.

(c) A doubling in a cars speed requires four times more centripetal force to act to keep the car on the road. Inexperienced drivers often fi nd this out when they run off the road into a tree or into another vehicle. Barely a day goes by without a news story about such crashes, caused by this quirk of physics.



Example 24

Find the maximum speed which a 1.5 tonne car can negotiate a curved section of road that has a radius of curvature of 200 m if the tyres can exert a maximum sideways force on the road of 3.0 kN before beginning to slide?

SolutionUse:

Fmv

rmv Fr

vFr

m

=

=

=

=

=

2

2

3

3

3 0 10 200

1 5 10

20

max

.

.

mm s1

The maximum speed of the car in such a curve is 20 m s1. Once the car begins to slide the maximum friction between the tyres and the road is reduced signifi cantly.



Example 25

Find the kinetic energy of a 250 kg motorbike moving at 30.0 m s1.

Solution

E mvk =

=

=

1

21

2250 30

1 13 10

2

2

5. J



Example 26

A net force of 800 N acts on the motorbike in Example 25 for a distance of 100 m.(a) Find the work done by this net force:(b) hence fi nd the motorbikes new speed.

Solution

(a) W = Fs = 800 N 100 m = 8.00 104 J(b) E

k = 1.13 105 J + 8.00 104 J

= 1.93 105 J

To fi nd the bikes new speed:

E mv

vE

m

k

k

=

=

=

=

1

2

2

2 1 93 10

250

39 3

2

5.

. m s1



Example 27

The brakes of a car or truck transform kinetic energy into heat. Assume that this transformation is 100% complete.(a) Find the amount of heat energy produced by the brakes when a 40 tonne truck is

stopped from a speed of 25 m s1; and(b) Find the distance over which a force of 2.0 104 N must act in order to do the

same amount of work as the brakes in part (a).

Solution(a) The amount of heat energy produced by the brakes equals the kinetic energy of

the truck:

Ek = 1

2 mv2

= 12 40 103 252

= 1.3 107 J or 13 MJ

Therefore, 12.5 MJ of heat is produced by the brakes of the truck.

(b) Work done = change in Ek

Fs = 1.25 107 J and since F = 2.0 104 N

m or 630 m

s =

=

1 25 10

2 0 10

6 3 10

7

4

2

.

.

.



Example 28

If the truck in Example 27 is stopped in a distance of 3.0 m what average force must be exerted?

SolutionThe same equation, W = Fs can be used with s = 3.0 m, W = 1.25 107 J:

FW

s=

=

=

1 25 10

3 0

4 2 10

7

6

.

.

. N



Example 29

(a) Find the momentum of the truck in Example 27.(b) Find the impulse on the truck when it is stopped.(c) Using equations for momentum and impulse, fi nd the time taken to stop the truck

using your answer from Example 28.

Solution

(a) p mv=

=

=

40 10 kg 25 m s

kg m s

3 1

11 0 106.

(b) Use I = p and since the truck is stopped, I = pi

= 1.0 106 N s

(c) Use:

N s

N

I Ft

tI

F

s

=

=

=

=

1 0 10

4 2 10

0 24

6

6

.

.

.

This would represent the conditions found in a severe crash into an object such as another large truck.



Example 30

On the same axes, sketch a graph showing the relationship between:(a) an objects momentum as its speed varies; and(b) an objects kinetic energy as its speed varies.

Solution

The graph shows how momentum is proportional to the speed of the object, while kinetic energy is proportional to the square of the speed of the object.

0 speed

kinetic energy

momentum



Example 31

A rocket motor exerts a force of 45 kN for 20 s. What is the impulse given by the motor?

Solution

Use: I = Ft = 45 103 N 20 s = 9.0 105 N s



Example 32

Explain how a future spaceship motor that exerts a small force over a long time could produce the same increase in speed as a motor exerting a large force over a short time.

SolutionOnce in space there is no friction acting against the motion of a spaceship. A small but constant force acting over a long time can impart the same impulse as a larger force acting for a short time, thus increasing the speed of the spaceship. Such propulsion systems have been proposed for future space travel and exploration.

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