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Question: A police boat is chasing a boat with criminals along a straight river by moving against the stream. The speed of the river stream is 3 miles per hour, the speed of the boat with criminals relative to the river is 30 miles per hour, and the police boat is 4 miles per hour faster than the boat with criminals.

Currently the criminals are ahead of the police, and horizontally throw a stone at the police boat at a speed 16 miles per hour relative to their boat (i.e. relative to the boat of criminals).

What is the horizontal velocity of the stone relative to the police boat and to the river bank? You need to state what the origin and the positive direction of motion are.

Choose the direction of the river stream to be the positive direction and choose the origin to be in front of criminals' boat.

Denote the velocity of the river stream by $\dot{x}_R$ Denote the velocity of the police boat by $\dot{x}_P$ Denote the velocity of the criminal boat by $\dot{x}_C$. Denote the velocity of the stone thrown by $\dot{x}_S$.

From the question we have that $\dot{x}_R = 3$, and we also have that the velocity of the criminal boat relative to the river stream is \begin{align*} \dot{x}_C - \dot{x}_R &= -30 \\ \implies \dot{x}_C - 3 &= -30 \\ \implies \dot{x}_C &= -27 \end{align*} Note that the minus 30 is because from the point of view of the river, the criminal boat is travelling in the negative direction.

Now from $\dot{x}_C$ I can calculate $\dot{x}_P$, since \begin{align*} \dot{x}_P &= \dot{x}_C + (-4) \\ &= -27 - 4 \\ &= -31 \end{align*}

Note that the minus 4 is because the police boat is 4 mph faster, but in the negative direction.

Also from $\dot{x}_C$ I can calculate $\dot{x}_S$. From the question we have that \begin{align*} \dot{x}_S - \dot{x}_C &= 16 \\ \implies \dot{x}_S - (-27) &= 16 \\ \implies \dot{x}_S &= -11 \end{align*}

I was wondering if my solution was right, even though $\dot{x}_S = -11$ is negative, even though it moves in the positive direction.

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closed as off-topic by Brandon Enright, John Rennie, David Z Nov 27 '13 at 18:33

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All correct. Your values for $ \dot{x}_P$ and $ \dot{x}_C$ are negative because you defined $ \dot{x}_R$ to be positive (both boats are moving against the river). No problem there, this is completely up to you, but to satisfy the requirement of the question you are required to state that motion of the river has been designated by you as positive direction (equivalent to stating that the boat movement direction are designated negative direction).

Also by defining $ \dot{x}_R$=+3, this implies a reference being made to the river bank (the origin). Your subsequent equations involving this definition of $ \dot{x}_R$ therefore also produce velocities with reference to the river bank. so the values -27, -31, -11 are all velocities relative to the river bank.

You did not obtain the velocity of the stone relative to the police boat. This is $ \dot{x}_S- \dot{x}_P$, using values you already obtained.

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  • $\begingroup$ Why have you chosen the river bank as the origin? $\endgroup$ – Michael Nov 27 '13 at 18:52
  • $\begingroup$ "The speed of the river stream is 3 miles per hour." it has to be 3mph relative to something. not exactly obvious, but it should be taken that this measurement is relative to the river bank, i.e, water is flowing along the river bank at 3mph. $\endgroup$ – gregsan Nov 27 '13 at 18:57
  • $\begingroup$ I forgot to include this in question, but is $\dot{x}_S - \dot{x}_P = 20$ mph and $\dot{x}_S - \dot{x}_{RiverBank}$ = -11 mph? Thanks for explanation on the origin. $\endgroup$ – Michael Nov 27 '13 at 19:02
  • $\begingroup$ both correct. v_riverbank=0 since it is the origin. it is good practice to write v_police-v_bank=-31 rather than v_police=-31, because the former implicitly shows the use of the riverbank as reference point. otherwise you have to include a statement that says all your velocity values are relative to some point (as the question asks). $\endgroup$ – gregsan Nov 27 '13 at 19:07
  • $\begingroup$ "otherwise you have to include a statement that says all your velocity values are relative to some point", does this just mean I have to state that the origin is the river bank? $\endgroup$ – Michael Nov 27 '13 at 19:31
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Note that the speed of the criminals relative to the river ($27\,\mathrm{mph}$) is larger than the speed of the stone relative to them ($11\,\mathrm{mph}$). Then, the resulting velocity would be expected to be in the negative direction

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  • $\begingroup$ Oh ok, so since it's larger, then the resulting velocity is to be in the negative direction. That clears that up. Thanks! $\endgroup$ – Michael Nov 27 '13 at 18:53

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