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 16 knots with heading 190°. Find the resulting speed and direction of the barge.
I can do the problems involving one force acting, but the two forces acting are throwing me off...would someone care to help please?
Thank you for your time!
As mathguy has helpfully pointed out, you need to resolve both velocities into the $x$ and $y$ components.
The figure below should help you to resolve the velocities.
By trigonometry (using the convention that East/right is positive), the overall horizontal component is given by $$x_v=15\cos(40^\circ)-16\cos(80^\circ)=8.7123$$ So the barge is heading in an easterly direction.
Using the convention that North/up is positive, the overall vertical component is given by $$y_v=-15\sin(40^\circ)-16\sin(80^\circ)=-25.399$$ So the barge is heading in a southerly direction.
Using Pythagoras, the resultant speed of the barge is given by $$\sqrt{8.7123^2+(-25.399)^2}=26.851$$
As for the direction in which the barge is travelling, the clockwise angle with respect to the East axis will be $$\arctan\left(\frac{25.399}{8.7123}\right)=71.1^\circ$$ resulting in a heading of $161.1^\circ$