# Relative velocities of a boat and a stone thrown by the boat.

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.

Your solution is correct and tells you that the stone does not move in the positive direction. Relative to the riverbank, the stone is moving upstream at $11$ miles per hour.