(hrwc9p75_6e) You are on an iceboat on frictionless, flat ice; you and the boat
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Question
(hrwc9p75_6e) 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 ml and m2 such that M = 6.0000m1 =12.000m2. To get the boat moving, you throw the stones rearward, either in succession or together, but in each case with a certain speed 12.000 m/s relative to the boat. What is the resulting speed of the boat if you throw the stones simultaneously? What is the resulting speed of the boat if you throw the stones m1 and then m2? What is the resulting speed of the boat if you throw the stones m2 and then m1?Explanation / Answer
Boat and person mass = M
stones
m1 = M/6 kg
m2 = M/12 kg
Velocity of stones relative to boat= v = 12 m/s
Velocity of boat: Vb.
a) M * Vb = (M / 6 + M / 12) *v
Vb = (1 / 6 + 1 / 12) v = v / 4
Vb = v / 4
Vb = 12 / 4 = 3 m/s
the speed of the boat when the stone thrown simultaniously is 3 m/s
b)
After throwing first stone, m1. Velocity of boat: Vb1.
(M + m2) * Vb1 = m1 *v = M * v / 6
(M + M / 12) Vb1 = M *v/6
(13/12) * Vb1 = v/6
Vb1 = v / 6 (12 / 13) = ( 2 / 13) * v
Vb1 = (2* 12 / 13) = 1.84 m/s
After throwing second stone, m2. Velocity of boat: Vb2.
M (Vb2 - Vb1) = m2 *v
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
Vb = 37*v / 156 = 37 * 12 / 156
Vb= 2.844 m/s
the resulting speed of the boat if you throw the stones m1 and then m2 is 2.844 m/s
c) After throwing first stone, m2. Velocity of boat: Vb1.
(M + m1) Vb1 = m2 *v = M *v/12
(M + M / 6) Vb1 = M *v/12
(7 / 6) Vb1 = v / 12
Vb1 = v / 12 *(6/7) = v *(1/12) * (6/7) = 12 * (1/14)
Vb1 = 0.857 m/s
After throwing second stone, m1. Velocity of boat: Vb2.
M *(Vb2 - Vb1) = m1 *v
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
Vb = 5v/21 = 5*12 / 21 = 2.856 m/s
the resulting speed of the boat if you throw the stones m2 and then m1 is 2.856 m/s
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