# At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per h

Nov 10, 2017

${v}_{A B} = \setminus \sqrt{{16}^{2} + {21}^{2}} = 26.401$ knots.

#### Explanation:

${\vec{v}}_{A B}$ : Velocity of ship A relative to ship B,

${\vec{v}}_{A S}$ : Velocity of ship A relative to sea,

${\vec{v}}_{B S}$ : Velocity of ship B relative to sea,

Consider a cartesian coordinate system in which X axis is oriented East and Y axis is oriented North.

${\vec{v}}_{A S} = \left(- 16 , 0\right)$ knots;

${\vec{v}}_{B S} = \left(0 , + 21\right)$ knots;

${\vec{v}}_{A B} = {\vec{v}}_{A S} + {\vec{v}}_{S B} = {\vec{v}}_{A S} - {\vec{v}}_{B S}$

$\setminus {\vec{v}}_{A B} = \left(- 16 , 0\right) - \left(0 , 21\right) = \left(- 16 , - 21\right)$ knots

${v}_{A B} = \setminus \sqrt{{16}^{2} + {21}^{2}} = 26.004$ knots

"at 4 pm" is just a red herring.