AP Physics

Semester Review26 is torque

32, 45-52, 58, 59 deal with momentum

1. The position of a particle moving along the x axis is given by , where t is in seconds. What is the average velocity during the time interval t = 1.0 s to t = 3.0 s?

x 21 22t 6t 2 m

x x3 x1

x 21 22 3 6 3 2 21 22 1 6 1 2 mx 33 37 mx 4m

v xt

4m2sec

2 msec

2

3

msec

2. A bullet is fired through a board, 50.0 cm thick, with its line of motion perpendicular to the face of the board. If it enters with a speed of 500 and emerges with a speed of 200 , what is the bullet's acceleration as it passes through the board?

msec

v f2 v i

2 2ax

v f2 v i

2

2xa

200 msec 2 500 m

sec 2

2 0.5m a 210000 m

sec2

4

3. The position of a particle moving along the x axis is given by x = 6.0t2 - 1.0t3, where x is in meters and t in seconds. What is the position of the particle when it achieves its maximum speed in the positive x direction?

v dxdt

12t 3t 2

a dvdt

12 6t

0 12 6tt 2sec

x 6.0t 2 1.0t 3

x 6.0 2 2 1.0 2 3

x 24 8 mx 16m

5

4. A particle moving along the x axis has a position given by x = (54t - 2t3)m, where t is in seconds. How far is the particle from the origin (x = 0) when the particle is not moving?

v dxdt

54 6t 2

0 54 6t 2

t 3sec

x 54t 2.0t 3

x 54 3 2.0 3 3

x 162 54 mx 108m

The particle is not moving at the instand the velocity equals zero.

6

vo

v 14v

vTop 0

34 H

v f2 v 1

4

2 2g 34 H

0 v 14

2 2g 34 H

0 v 2 g 32 H

2 v 2 3g

H

5. An object is thrown vertically and has an upward velocity of v when it reaches one fourth of its maximum height above its launch point. What is the initial (launch) speed of the object?

v f2 vo

2 2gH

vo 2gH

vo 2g 2v 2

3g

vo 43 v 2

vo 43v

7

6. A ball thrown vertically from ground level is caught 3.0 seconds later by a person on a balcony which is 30 meters above the ground. Determine the initial speed of the ball.

y v iyt 12 gt 2

y 12 gt 2

tv iy

30 12 10 m

sec2 3sec 2

3sec v iy 25 m

sec

8

7. Vx is the velocity of a particle moving along the x axis as shown. If x = 2.0 m at t = 1.0 s, what is the position of the particle at t = 6.0 s?

1 2 3 4 5 6

-2

-1

0

1

2

3

4

t sec

vxmsec

∆x is the area under the line. The area between t = 1 sec and t = 3 sec cancels out. So we’ll find the area from t = 3 sec to t = 6 sec.

x 3 2 12 3m

2m 3m 1m

9

8. An object is thrown vertically upward such that it has a speed of 25 when it reaches two thirds of its maximum height above the launch point. Determine this maximum height.

msec

vo

v 2325 m

sec

vTop 0

23 H

13 H

vTop2 v 2

3

2 2g 13 H

3v 23

2 2gH

3 v 23

2 2g

H 3 25 m

sec 2

2 10 msec2

H 93.75m

10

9. Object A is thrown downward at t = 0 with an initial speed of 10 from a height of 60m above the ground. At the same instant, a object B is propelled vertically upward from ground level with a speed of 40 . At what height above the ground will the two objects pass each other?

msec

msec

H 60 vA t 12 gt 2 H vB t 1

2 gt 2

60 vA t 12 gt 2 vB t 1

2 gt 2

60 vA t 12 gt 2 vB t 1

2 gt 2

60 10t 40t 60 50t 1.2sec t

10. A car travels north at 30 for one half hour. It then travels south at 40 for 15 minutes. The total distance the car has traveled and its displacement are:

11

msec

msec

x total 30 msec 30min 60sec

min

40 m

sec 15min 60secmin

90km

x 54 ˆ j km 36 ˆ j km 18km, north .

12

11. A particle starts from the origin at t = 0 with a velocity of 8.0i and moves in the xy plane with a constant acceleration of (-2.0i + 4.0j) . At the instant the particle achieves its maximum positive x coordinate, how far is it from the origin?

y vyit 12 a ˆ j t

2

y 12 4 ˆ j m

sec2 t 2

y 2t 2

y 32m

x 8t t 2

v dxdt

8 2t

0 8 2tt 4 sec

x 8t t 2

x 8 4 4 2

x 16m

r i2 j 2

r 35.8m

13

12. A rock is projected from the edge of the top of a 100-ft tall building at some unknown angle above the horizontal. The rock strikes the ground a horizontal distance of 160 ft from the base of the building 5.0 seconds after being projected. Assume that the ground is level and that the side of the building is vertical. Determine the speed with which the rock was projected.

vx xt

160 ft5sec

32 ftsec

y vyit 12 gt 2

12 gt 2 y vyit12 gt 2 y

tvyi

16 ftsec2 5sec 2 100 ft

5secvyi

60 ftsec vyi

v vxi2 vyi

2

v 68 ftsec

14

13. A football is thrown upward at a 30° angle to the horizontal. To throw a 40.0-meter pass, what must be the initial speed of the ball?

R vo2 sin2

gRg

sin2vo

2

vo Rg

sin2

40m 10 msec2

sin 60vo 21.5 m

sec

14. The horizontal surface on which the objects slide is frictionless. If M = 2.0 Kg and the Tension in string 1 is 18 Newtons, Determine F.

15

FNet maT1 3MaT1

3Ma

18N6Kg

a 3 msec2

F 5Ma

F 5 2Kg 3 msec2

F 30N

16

15. A particle starts from the origin at t = 0 with a velocity of and moves in the xy plane with a constant acceleration of . At the instant the x coordinate of the particle is 32 meters, what is the magnitude of the y coordinate?

a 4i 2 j msec 2

8 j msec

x 12 ait

2

32m 12 4 m

sec2 t 2

t 4 sec

y vyit 12 a j t

2

y 8 msec 4 sec 1

2 2 msec2 4 sec 2

y 48m

16. The surface of an inclined plane is frictionless. If F = 30N, what is the magnitude of the force exerted on the 3kg block by the 2kg block

FNet maF FP m1 m2 aF m1 m2 gsin m1 m2 a30N 5Kg 9.8 N

Kg 0.5 5Kg

a

30N 24.5N5Kg

a 1.1 msec2

FP

17

3Kg

Push

FP

FNet maP FP maP ma FP

P ma mgsin P 3Kg 1.1 m

sec2 3Kg 9.8 NKg 0.5

P 3.3N 14.7NP 18N

18

2 Kg 3 Kg 5 KgP 1 2 3

17. If P = 5.0 N, what is the magnitude of the force exerted on block 1 by block 2?

First, find the acceleration of the system!

a FNet

m

Pm

a 5N10Kg

0.5 msec2

19

P Push maP ma Push

5N 2Kg 0.5 msec2 Push

Push 5N 1NPush 4N

Push ma

Push 8Kg 0.5 msec2

Push 4N

P

Push

2Kg

Push

3Kg

5Kg

For every action Force there is an equal and opposite reaction Force!

20

21

y

x

20

b

30°

18. If a = (20i + 2.7j), and b is as shown, what is the magnitude of the Resultant of the two vectors?

20 cos120 10i

20 sin120 17.3 j

b 10i 17.3 j

a b 10i 20 j a b 102 202

a b 22.4

19. The graph below shows the force on an object of mass m as a function of time. For the time interval t = 0 to t = 3 seconds, the total change in the momentum of the object is

p Fdtp 20N sec 10N sec p 10N sec

22

20. The position of a particle moving along the x axis is given by , where x is in meters and t is in seconds. What is the average acceleration between 2 sec < t < 4 sec ?

23

x 2t 3 6 t2 4

v dxdt

6t 2 12t

a dvdt

12t 12

a ai a f

2

a 12 36 m

sec2

2a 24 m

sec2

21. If M = 5.0 Kg, what is the Tension in the string between the two objects of equal mass? All surfaces are frictionless.

24

3M

M

M

T

a 3M 2M3M 2M

g

a M5M

g 15 g

25

FNet maT Mg MaT Ma MgT M a g T 5 1

5 g 55 g

T 5 65 g

T 60N

M

T

Mg

22. If the tension T = 15 N and the acceleration , what is the mass m of the suspended object? All surfaces are frictionless!

26

a 5 msec2

Ta

m

m

T

mg

FNet mamg T mamg ma Tm g a T

m Tg a

m 15N5 m

sec2

3Kg

27

1 2

m

60°

23. If the Tension in string 1 is 40N and the Tension in string 2 is 25N, what is mass m of the object?

2

mg

30

Z

25Nsin 30

40Nsin Z

Z 53

180 83 97 D

25Nsin 30

Wsin 97

W 49.6N

m Wg

49.6N10 N

kg4.96kg

D

28

24. A roller-coaster car has a mass of 500kg when fully loaded with passengers. The car passes over a hill of radius 20 m. At the top of the hill, the car has a speed of 8.0 . What is the force of the track on the car at the top of the hill?

msec

mg

N

FC mg N

N mg mv2

R

N 500kg 10 Nkg 500kg 8 m

sec 2

20mN 3400N, up

25. A uniform ladder, 15 ft long and weighing 60 lbs., is leaning against a frictionless wall at an angle of 53° above the horizontal. A 120 lb boy climbs 6.0 ft up the ladder. What is the magnitude of the friction force exerted on the ladder by the floor?

60lbs

Nwall

Nground

f

60lbscos

f cos f sin

53

120lbs29

CW CCW

60lbs cos53 7.5 ft 120lbs cos53 6 ft f cos37 15 ft 60lbs 3

5 7.5 120lbs 35 6 f 4

5 15 270lbs 432lbs f 12

702 lbs12

f 58.5 lbs

30

31

26. A uniform, horizontal meter stick, supported at the 50 cm mark, has a mass of 500 grams hanging at the 20 cm mark and a 300 gram mass hanging at the 60 cm mark. Determine the position on the meter stick at which one would hang a third mass of 600 grams to keep the meter stick balanced.

500g

300g

600g

d

CCW CW

500g g 30cm 300g g 10cm 600g g d 5 30cm 3 10cm 6 d 150cm 30cm 6d

120cm6

d 20cm the 70cm mark!

32

27. The three blocks are released from rest. Determine the acceleration of the system if no friction is present.

M

2M

3M

T2

T2

T1

T1

3Mg

Mg

FNet ma3Mg Mg 6Ma 3g g 6a

g3

a

33

28. A person weighing 600N rides in an elevator that has a downward acceleration of 0.75 . What is the magnitude of the force of the elevator floor on the person?

msec 2

mg

N

FNet mamg N mamg ma Nm g a N

60kg 9.25 Nkg N

555N N

34

29. A 0.20-kg object attached to the end of a string swings in a vertical circle of radius = 80 cm. At the top of the circle the speed of the object is 4.5 . What is the magnitude of the tension in the string at this position?

v

T

mg

mg T FC

T FC mg

T m v 2

R g

T 0.2kg4.5 m

sec 2

0.8m 10 m

sec2

T 3.06N

30. A roller-coaster car has a mass of 500 kg when fully loaded with passengers. At the bottom of a circular dip of radius 40 m (as shown in the figure) the car has a speed of 25 . What is the magnitude of the force of the track on the car at the bottom of the dip?

35

msec

25 msec

N mg FC

N FC mg

N m v 2

R g

N 500kg25 m

sec 2

40m10 m

sec2

N 12812.5N

40m

31. Daniel’s apparent weight at the top of an inverted loop of radius 30 m while riding a roller coaster is 3W. What is Daniel’s apparent weight at the bottom of the loop?

36

W

3W

N

W

At the top!

FC W 3W 4W

At the bottom!

FC N WFC W N4W W N5W N

32. As shown in the top view above, a disc of mass m is moving horizontally to the right with speed v on a table with negligible friction when it collides with a second disc of mass 2m. The second disc is moving horizontally to the right with speed v/2 at the moment of impact. The two discs stick together upon impact. The speed of the composite body immediately after the collision is

37

pi p f

mv 2m v2 p f 2mv

v p f

m

2mv3m

23 v

38

33. A 0.30-kg mass attached to the end of a string swings in a vertical circle (R = 1.4 m), as shown. At an instant when = 30°, the speed of the mass is 6.0. What is the magnitude of the resultant force on the mass at this instant?

R

v

m

g

FC mv2

R

FC 0.3kg 6 m

sec 2

1.4mFC 7.71N

39

34. A race car traveling at 100 enters an unbanked turn of 400 m radius. The coefficient of friction between the tires and the track is 1.1. The track has both an inner and an outer wall. Which statement is correct?

FC mv2

Rmg

v Rg 1.1 400m 10 msec2

v 66.3 msec

The driver and car crash into the outer wall!

msec

40

35. A force acting on an object moving along the x axis is given by where x is in meters. If the force is applied at an angle of 37° with respect to the horizontal, how much work is done by this force as the object moves from x = -1 m to x = +2 m?

Fx 14x 3x 2 N

W F x Fxcos

W F x dx

W 14x 3x 2 1

2 dx

W 7x 2 x 3 1

2cos

W 28 8 7 1 45

W 12 45 9.6J

41

36. A particle of mass m moves along a straight path with a speed v defined by the function , where b and c are constants and t is time. What is the magnitude F of the net force on the particle at time t = t1 ?

v bt 2 c

a dvdt

2bt

F maF 2bmt1

42

37. A 0.50-kg particle moves under the influence of a single conservative force. At point A where the particle has a speed of 10 , the potential energy associated with the conservative force is +40 J. As the particle moves from A to B, the force does +25 Joules of work on the particle. What is the value of the potential energy at point B?

msec

K i 12 mvi

2

K i 12 0.5kg 10 m

sec 2

K i 25J

ET K U 65J

If the KE increases by +25J, then the U must decrease by the same amount.

U 40J 25J 15J

38. To stretch a certain nonlinear spring by an amount x requires a force F given by , where F is in newtons and x is in meters. What is the change in potential energy when the spring is stretched 2 meters from its equilibrium position?

F 40x 6x 2

U F x dx

U 40x 6x 2 0

2 dx

U 20x 2 2x 3 0

2

U 80 16 U 64J

43

39. A skier weighing 550N comes down a frictionless ski run that is circular (R = 30 m) at the bottom, as shown. If her speed is 12 at point A, what is her speed at the bottom of the hill (point B)?

msec

EA EB

KA UA KB

12 mvA

2 mgh 12 mvB

2

vA2 2gh vB

2

vA2 2gh vB

12 msec 2 19.6 m

sec2 7.02m vB

vB 16.8 msec

h R Rcos 7.02m

44

40. A block (mass = 4.0 kg) sliding on a horizontal frictionless surface is attached to one end of a horizontal spring ( k =100 ) which has its other end fixed. If the maximum distance the block slides from the equilibrium position is equal to 20 cm, what is the speed of the block at an instant when it is a distance of 16 cm from the equilibrium position?

Nm

US U16cm K16cm

12 kA2 1

2 kx 2 12 mv2

kA2 kx 2 mv2

k A2 x 2 mv2

v km A2 x 2

v 100 Nm

4kg 0.2m 2 0.16m 2 v 0.6 m

sec

45

41. A 1.2-kg mass is projected down a rough circular track (radius = 2.0 m) as shown. The speed of the mass at point A is 3.2 , and at point B, it is 6.0 . How much work is done on the mass between A and B by the force of friction?

msec

msec

A

B

R

KA UA W f KB

KA UA KB W f

12 mvA

2 mgR 12 mvB

2 W f

6.144J 23.52J 21.6J W f

8.06J W f

46

42. A 25-kg block on a horizontal surface is attached to a light spring ( k = 8.0 ). The block is pulled 10 cm to the right from its equilibrium position and released from rest. When the block has moved 5.0 cm toward its equilibrium position, its kinetic energy is 12 J. How much work is done by the frictional force on the block as it moves the 5.0 cm?

kNm

U10cm W f U8cm K

U10cm U8cm K W f

12 kA2 1

2 kx 2 K W f

12 k A2 x 2 K W f

8000 Nm

20.1m 2 0.05m 2 12J W f

30J 12J W f 18J47

43. As an object moves from point A to point B only two forces act on it: one force is nonconservative and does -30 Joules of work, the other force is conservative and does +50 Joules of work. Between A and B,

48

44. A 2-kg object moving with a speed of 6.0 collides perpendicularly with a wall and emerges with a speed of 6.0 in the opposite direction. If the object is in contact with the wall for 2.0 msec, what is the magnitude of the average force on the object by the wall?

msec

msec

F pt

F m v f v i

t

F 2kg 6 m

sec 6 msec

0.002secF 12000N

49

45. A 1.0-kg playground ball is moving with a velocity of 6.0 directed 30° below the horizontal just before it strikes a horizontal surface. The ball leaves this surface 0.50 s later with a velocity of 4.0 directed 60° above the horizontal. What is the magnitude of the average resultant force on the ball?

msec

msec

v i 5.2ˆ i 3ˆ j msec v f 2ˆ i 3.46 ˆ j m

sec

v v f v i

v 2ˆ i 3.46 ˆ j msec 5.2ˆ i 3ˆ j m

sec

v 3.2ˆ i 6.46 ˆ j msec

v ˆ i 2 ˆ j 2 7.2 msec

50

Ft mv

F mvt

F 1.0kg 7.2 m

sec 0.5sec

F 14.4N

51

46. The only force acting on a 2.0-kg object moving along the x axis is shown. If the velocity vx = -2.0 at t = 0, what is the velocity at t = 4.0 seconds?

msec

Fx (N)

t sec

1

2

3

4

8

4

p Fdt 4 2 8 kgmsec

p 2 kgmsec

p p f pi

p pi p f

2 kgmsec 4 kgm

sec p f 6 kgmsec

v f p f

m

6 kgmsec

2kg 3 m

sec

52

47. The speed of a 2.0-kg object changes from 30 to 40 during a 5.0-second time interval. During this same time interval, the velocity of the object changes its direction by 90¯. What is the magnitude of the average total force acting on the object during this time interval?

msec

msec

v v f v i

v 0ˆ i 40 ˆ j msec 30ˆ i 0 ˆ j m

sec

v 30ˆ i 40 ˆ j msec

v ˆ i 2 ˆ j 2 50 msec

Ft mv

F mvt

F 2kg 50 m

sec 5sec

F 20N

53

48. A 2.0-kg object moving 5.0 collides with and sticks to an 8.0-kg object initially at rest. Determine the kinetic energy lost by the system as a result of this collision.

msec

pi p f

mv m M v

mm M

v v

2kg10kg

5 msec v

1 msec v

K K i K f

K 12 mvi

2 12 m M v f

2

K 12 2kg 5 m

sec 2 12 10kg 1 m

sec 2

K 25J 5JK 20J

54

49. A 2-kg block is attached to the end of a 3.0-m string to form a pendulum. The pendulum is released from rest when the string is horizontal. At the lowest point of its swing when it is moving horizontally, the block is hit by a 25-gram bullet moving horizontally in the opposite direction. The bullet remains embedded in the block and causes the block to come to rest at the low point of its swing. What was the magnitude of the bullet's velocity just before hitting the block?

t

h

u

d

v 0

3m

2kg

2kg

vBi 2ghvBi 7.75 m

sec

vbi

55

pi p f

pBi pbi 0 pBi pbi

mBvBi mbvbi

mBvBi

mb

vbi 2Kg 7.75 m

sec 0.025kg

vbi 620 msec

56

50. A 3.0-kg mass sliding on a frictionless surface has a velocity of 5.0 east when it undergoes a one-dimensional inelastic collision with a 2.0-kg mass that has an initial velocity of 2.0 west. After the collision the 3.0-kg mass has a velocity of 1.0 east. How much kinetic energy does the two-mass system lose during the collision?

msec

msec

msec

t

h

u

d

3kg

2kg

5.0 msec

2kg

3kg

2 msec

1 msec

15 kgmsec

4 kgmsec

4 kgmsec

7 kgmsec

v pm

7 kgm

sec

2kgv 3.5 m

sec

57

K i 12 m1v1

2 12 m2v2

2

K i 12 3kg 5 m

sec 2 12 2kg 2 m

sec 2

K i 37.5J 4J 41.5J

KLost K i K f 26.75J

K i 12 m1 v 1

2 12 m2 v 2

2

K f 12 5Kg 1 m

sec 2 12 2Kg 3.5 m

sec 2

K f 2.5J 12.25J

K f 14.75J

51. A 80-kg man who is ice skating south collides with a 40-kg boy who is ice skating east. Immediately after the collision, the man and boy are observed to be moving together with a velocity of 2.0 , in a direction 37¯ south of east. What was the magnitude of the boy's velocity before the collision?

msec

t

h

u

d

37

40kg

80kg

p 240 kgmsec , at 37

p 192ˆ i 144 ˆ j kgmsec

80kg

40kg

vboy pm

vboy 192 Kgm

sec

40kgvboy 4.8 m

sec

59

52. A 6-kg object, initially at rest, "explodes" into three objects of equal mass. Two of these are determined to have velocities of equal magnitudes (4.0 ) with directions that differ by 90¯. How much kinetic energy was released in the explosion?

msec

b

o

o

m

4 msec

6kg

90

4 msec

2kg

0

0

H

V

8 ˆ j kgmsec

8ˆ i kgmsec

8ˆ i kgmsec

8 ˆ j kgmsec

2kg

2kg

p ˆ i 2 ˆ j 2

p 82 82

p 11.3 Kgmsec

v pm

v 11.3 Kgm

sec

2.0kgv 5.66 m

sec

60

K 12 mv2 16J

K 12 mv2 16J

K 12 mv2 32J

KNet KKNet 16J 16J 32JKNet 64J

61

53. A particle moves in the xy plane with a constant acceleration given by . At t = 0 its position and velocity are 10i m and , respectively. What is the distance from the origin to the particle at t = 2.0 s?

a 4.0 j msec2

2i 8 j msec

y vyit 12 ayt

2

y 8 msec 2sec 1

2 4 msec2 2sec 2

y 8 ˆ j m

ˆ r 6i 8 j mˆ r ˆ i 2 ˆ j 2 10.0m

62

54. A ball is thrown horizontally from the top of a building 80 meters high. The ball strikes the ground at a point 100 meters horizontally away from and below the point of release. What is the speed of the ball just before it strikes the ground?

t 2yg

t 2 80m 10 m

sec2 t 4.0sec

vx xt

100m4sec

25 msec

vy gt 40 msec

v vx2 vy

2 47.2 msec

63

0

10

20

-10

30F(N)

x (m)2 4 6 8 10 12

A 5-kg mass starts from rest under the influence of variable force F as a function of distance x as shown on the graph below.

55. During which time interval(s) will the velocity of the mass be constant?

When the acceleration = 0 and the Force = 0. Between x = 6 m and x = 8 m.

64

0

10

20

-10

30F(N)

x (m)2 4 6 8 10 12

56. During which time interval will the most Work be done?

W F x dxThe areas between x = 0 and x 4 m, between x = 4 m and x = 6 m, and between x = 8m and x = 12 m are all the same. So the same work is done in all three areas.

65

57. What is the velocity of the object at x = 12 m?

0

10

20

-10

30F(N)

x (m)2 4 6 8 10 12

W K F x dx

K 40 40 40 JK 40J

K 12 mv f

2 12 mvi

2

K 12 mv f

2

v f 2K

m

2 40J 5kg

4 msec

66

An object of mass m, initially at rest, slides down from a height of 1.25 meters on a frictionless ramp, collides and sticks to an identicle particle 2 of mass m at rest as shown. Then particle 1 and 2 together collide elastically with particle 3 of mass 2m which is also initially at rest.

m

1

2

3

m

2m

1.25m

58. The speed of particle 1 after the collision with particle 2 would be?

Double the mass = half the velocity!

vo 2ghvo 5 m

sec

2.5 msec

67

m

1

2

3

m

2m

1.25m59. The speed of particle 1 and

2 together after the elastic collision with particle 3 would be?

v1 f m1 m2

m1 m2

v1i

v f 2m 2m2m 2m

2.5 msec

v f 0

68

60. The same force F is applied horizontally to bodies 1, 2, 3 and 4, of masses m, 2m, 3m and 4m, initially at rest and on a frictionless surface, until each body has traveled distance 2d. The correct listing of the magnitudes of the velocities of the bodies, v1, v2, v3, and v4 is

v f2 v i

2 2a2d

v f 4ad

a1 Fm

a2 F

2m a3

F3m

a4 F

4m

69

v1 f 2 Fm d

v2 f 2 F2m 2d 2 F

2m d2v2 f 2 2 F

2m d2v2 f 2 2F

2m d v1 f 2v2 f

v3 f 2 F3m 2d 2 F

3m d

v4 f 2 F4m 2d F

m d

v1 f 2 ad 2 Fm d

70

v3 f 2 F3m d

3v3 f 2 3 F3m d

3v3 f 2 3F3m d

3v3 f 2 Fm d v1 f

v4 f Fm d

2v4 f 2 Fm d

2v4 f v1 f

v1 f 2 Fm d

71

v1 f 2v2 f 3v3 f 2v4 f

72

61. A constant force F is applied to a body of mass m that initially is headed east at velocity vo until its velocity becomes 2vo. The total time of travel is 3t. The total distance the body travels in that time is

m

m

v vo

to

1t

2t

3t

m

v 0

x vot 12 at 2

v vo

v 2vo

74

a Fm

a 0 vo

t

vo

t

Fm

vo

t

vo Fm

t

x vot 12 at 2

x vo 3t 12 a 3t 2

x 3 Fm t 2 9

2Fm t 2

x 32

Fm t 2

x 32

Fm t 2 F

m t 2 Fm t 2

x 72

Fm t 2

a Fm

a 0 vo

t

vo

t

Fm

vo

t

vo Fm

t

v f2 v i

2 2ax

2 Fm t 2

Fm t 2

2ax

3 Fm t 2

2 Fm

32

Fm t 2

x 32

Fm t 2 F

m t 2 Fm t 2

x 72

Fm t 2

a Fm

a 0 vo

t

vo

t

Fm

vo

t

vo Fm

t

x vo 3t 12 a 3t 2

x 62

Fm t 2 9

2 Fm t 2

x 32

Fm t 2

x total 2x Fm t 2

75