Find the resulting speed and direction. Trig Problem involving resultant and vectors.

Question

A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 15 knots with heading 130°, and the second tugboat is traveling at a speed of 11 knots with heading 190°. Find the resulting speed and direction of the barge. (Round your answers to the nearest whole number.)

$$ |v+w|^2=15^2+11^2-2(15)(11)cos120 $$ $$ |v+w|^2=511 $$ $$ |v+w| = 22.60530911 = 23 $$ $$ sinB= 11sin(12)/22.60530911 $$ $$ SinB=0.421417792 $$ $$ B=25 $$ $$ 130+25=155 $$

I'm getting a speed of $23$ knots and a direction of $155°$. Is this correct?

Answer 1

Your answer is correct. \begin{align*} x &= 15 \cos 130^\circ + 11 \cos 190^\circ \approx -20.4746994 \text{ knots} \\ y &= 15 \sin 130^\circ + 11 \sin 190^\circ \approx 9.5805366 \text{ knots} \\ r &= \sqrt{x^2 + y^2} \approx 22.6 \text{ knots} \\ \theta &= 180^\circ + \arctan (y \; / \; x) \approx 154.9^\circ \end{align*}

Answer 2

The question, and the OP's answer,(and the up-voted, accepted answer!), have the two tug-boats and the barge all travelling in different directions at different speeds. Thus, they must be getting steadily farther and farther apart. How is this possible? What does the diagram of the relative positions of the two tugs and the barge look like? How long are the two tow-ropes, and what happens when the tugs run out of rope?

The question as stated is impossible to answer, as it does not describe vectors that can be added trigonometrically in any meaningful way

If the velocities of the two tugboats were replaced with forces, and we were asked to find the size and direction of the resultant force on the barge, then the work shown would be correct...