Is my book's answer on a cons. of momentum question incorrect?

Question

I am reading Halliday Resnick and Walker's 4th edition of fundamentals of physics, and stumbled upon the following problem:

"You are on an iceboat (with combined mass M) on frictionless, flat ice. Along with you are two stones with masses $m_1$ and $m_2$ such that $M = 6m_1 = 12m_2$ 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 (a) when the stones are thrown together?"

I approached the problem in this way:

$$P_i = 0$$ $$P_f = -Mv_b + (M/6+M/12)v_{rel} = 0$$ $$v_b = v_{rel}/4$$

However, the back of the book claims the answer is $v_b = .200v_{rel}$.

Am I missing something?

Answer 1

The phrase "combined mass" is misleading here. It means "mass of everything except the stones". So the final mass of the boat (and everything else still on it) is indeed $M$. What's important is that $v_{rel}$ is the speed of the rocks relative to the boat, not relative to the original rest frame.

So the solution should go as follows.

$$P_f=-Mv_b+(M/6+M/12)(v_{rel}-v_b)=0$$

$$\implies v_b=\frac{3}{15}v_{rel}=0.200v_{rel}$$

Answer 2

You've taken $v_{rel}$ to mean the speed of the stones relative to the initial motion of the boat. While it is supposed to mean the speed relative to the final motion of the boat. So we should instead have $$Mv_b-\frac{M}{4}(v_{rel}-v_b)=0$$