Physic C...............help me!..........plz?

Question

You are on an iceboat on frictionless, flat ice; you and the boat have a combined mass M. Along with you are two stones with masses m1 and m2 such that M = 6.00 m1 = 12.0 m2. To get the boat moving, you throw the stones rearward, either in succession or together, but in each case with a certain speed v(rel) relative to the boat. What is the resulting speed of the boat if you throw the stones

a) simultaneously

b) m1 and then m2

c) m2 and then m1

Answer 1

In the first case the two stones together have 1/4 the mass of the boat and the rider. So the boat will travel at 1/4 the speed of the rocks. In the other two cases the second stone will travel relative to the earth at a slower speed so the boat will travel at a slower speed.

a) 0.25v

b) 37v/156 = 0.237v

c) 5v/21 = 0.238v

Boat and person mass = M

stones

m1 = M/6 kg

m2 = M/12 kg

Velocity of stones relative to boat: v

Velocity of boat: Vb.

a) MVb = (M/6+M/12)v

Vb = (1/6 + 1/12)v = v/4

ANSWER: Vb = v/4 = 0.25v

check:

momentum of boat: Mv/4

momentum of stones: (m1 + m2)v = (M/6 M/12)v = Mv(1/6+1/12) = Mv/4

b) After throwing first stone, m1. Velocity of boat: Vb1.

(M+m2)Vb1 = m1v = Mv/6

(M+M/12)Vb1 = Mv/6

(13/12)Vb1 = v/6

Vb1 = v/6(12/13) = v(1/6)(12/13) = v(12/6)(1/13) = (2/13)v

After throwing second stone, m2. Velocity of boat: Vb2.

M(Vb2-Vb1) = m2v

M(Vb2-(2/13)v) = (M/12)v

Vb2-(2/13)v = v/12

Vb2 = v/12+(2/13)v = v(1/12+2/13) = v(13/156+24/156) = 37v/156

ANSWER: Vb = 37v/156 = 0.237v

check:

momentum of boat: 37Mv/156

momentum of stones: m1v + m2(v-Vb1) = (M/6)v + (M/12)(v-(2/13)v)

= Mv(1/6+(1/12)(11/13)) = Mv(26/156+11/156)=Mv(37/156)

c) After throwing first stone, m2. Velocity of boat: Vb1.

(M+m1)Vb1 = m2v = Mv/12

(M+M/6)Vb1 =  Mv/12

(7/6)Vb1 =  v/12

Vb1 =  v/12(6/7) = v(1/12)(6/7) = v(1/14)

After throwing second stone, m1. Velocity of boat: Vb2.

M(Vb2-Vb1) = m1v

M(Vb2-v/14) = (M/6)v

Vb2-v/14 = v/6

Vb2 = v/6+v/14 =v(1/6+1/14) =v(7/42+3/42) = 10v/42

ANSWER: Vb = 5v/21 = 0.238v

check:

momentum of boat: Mv(5/21)

momentum of stones: m2v + m1(v-Vb1)= (M/12)v + (M/6)(v-V/14)

=Mv(1/12+(1/6)(13/14)) =Mv(1/12+(1/6)(13/14))

=Mv(7+13)/(3*4*7) =Mv(20/84)=Mv(5/21)