Q09. Systems of Particles

1. At the same instant that a 0.50-kg ball is dropped from 25 m

above Earth, a second ball, with a mass of 0.25 kg, is thrown

straight upward from Earth's surface with an initial speed of

15 m/s. They move along nearby lines and pass each other

without colliding. At the end of 2.0 s the height above

Earth's surface of the center of mass of the two-ball system

is:

1. 2.9 m

2. 4.0 m

3. 5.0 m

4. 7.1 m

5. 10.4 m

0

0.5 2516.7

0.5 0.25

kg mY m

kg kg

0

0.25 15 /5.0 /

0.5 0.25

kg m sV m s

kg kg

2

0 0

12.0 2.0 7.06

2Y Y V s g s m

Initial height of CM:

Initial velocity of CM:

Height of CM after 2s :

25m

0.25 kg15 m/s

0.5 kg0.0 m/s

V0

CMg

2. A 320-N hunter gets a rope around a 640-N polar bear.

They are stationary, 10 m apart, on frictionless level ice.

When the hunter pulls the polar bear to him, the polar bear

will move:

1. 1.0 m

2. 3.3 m

3. 10 m

4. 12 m

5. 17 m

0

640 10 20

640 320 3

N mX m

N N

0

1010 3.3

3X m X m m

Initial postion of CM:( Hunter at coord origin)

No external horizontal forces position of CM not changed.

Displacement of bear :

10 m

320 N 640 N

X

CM

X0

3. Two 4.0-kg blocks are tied together with a compressed spring between them.

They are thrown from the ground with an initial velocity of 35 m/s, 45° above

the horizontal. At the highest point of the trajectory they become untied and

spring apart. About how far below the highest point is the center of mass of the

two-block system 2.0 s later, before either fragment has hit the ground?

1. 12 m

2. 20 m

3. 31 m

4. Can't tell because the velocities of the fragments are

not given.

5. Can't tell because the coordinates of the highest point

are not given.

35 m/s45

g

4 kg 4 kg

Distance of free falling in 2.0 s is 2219.8 2 19.6

2m s s m

4. Which one of the following statements is true?

1. the CM of an object must lie within the object

2. all the mass of an object is actually concentrated at its CM

3. the CM of an object cannot move if there is zero net force

on the object

4. the CM of a cylinder must lie on its axis

5. none of the above

1. the CM of an object must lie within the object

2. all the mass of an object is actually concentrated at its CM

3. the CM of an object cannot move if there is zero net force on the object

4. the CM of a cylinder must lie on its axis

5. none of the above

Fnet = 0 V = const

Not true if ( ) ≠ (+ )

5. A 3.0-kg and a 2.0-kg cart approach each other on a horizontal

air track. They collide and stick together. After the collision

their total kinetic energy is 40 J. The speed of their center of

mass is :

1. zero

2. 2.8 m/s

3. 4.0 m/s

4. 5.2 m/s

5. indeterminate

2140 3.0 2.0

2J kg kg V

4 /V m s

6. Object A strikes the stationary object B head-on in an elastic

collision. The mass of A is fixed, you may choose the mass of

B appropriately. Then :

1. for B to have the greatest recoil speed, choose mA = mB

2. for B to have the greatest recoil momentum, choose mB << mA

3. for B to have the greatest recoil kinetic energy, choose mB >> mA

4. for B to have the least recoil speed, choose mB = mA

5. for B to have the greatest recoil kinetic energy, choose mB = mA

1. for B to have the greatest recoil speed, choose mA = mB

2. for B to have the greatest recoil momentum, choose mB << mA

3. for B to have the greatest recoil kinetic energy, choose mB >> mA

4. for B to have the least recoil speed, choose mB = mA

5. for B to have the greatest recoil kinetic energy, choose mB = mA

Momentum conservation

Relative velocity reversal

0B

A A BA

mv v v

m

0A B Av v v

02 A AB

A B

m vv

m m

02

1 /A A

B BA B

m vm v

m m

2 22 0

2

1 2

2B A A

B B

A B

m m vm v

m m

Greatest when B Am m

Greatest when B Am m

greatest when B Am m

7. Blocks A and B are moving toward each other along the x axis.

A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a

mass of 4.0 kg and a velocity of –25 m/s. They suffer an

elastic collision and move off along the x axis. The kinetic

energy transferred from A to B during the collision is :

1. 0

2. 2500 J

3. 5000 J

4. 7500 J

5. 10000 J

Initial total momentum : 2.0 50 / 4.0 25 / 0kg m s kg m s

Momentum conservation vA and vB after collision simply reverse their sign.

KE’s are not changed.

2.0 kg50 m/s

5.0 kg25 m/s

50 m/s 25 m/s

8. Two objects, X and Y, are held at rest on a horizontal

frictionless surface and a spring is compressed between

them. The mass of X is 2/5 times the mass of Y.

Immediately after the spring is released, X has a kinetic

energy of 50 J and Y has a kinetic energy of :

1. 20 J

2. 8 J

3. 310 J

4. 125 J

5. 50 J

Initial total momentum = 0

2 21 1

2 2Y X

Y Y Y X X XX Y

v mK m v m v K

v m

Momentum conservation X X Y Ym v m v

250 20

5J J

X : m = 2/5

Y : M = 1

K = 10 J