Im not sure where to begin with this problem or how to set it up. I am aware it
ID: 2045660 • Letter: I
Question
Im not sure where to begin with this problem or how to set it up. I am aware it is dealing with vectors and two-dimension motion, but I am confused on how to begin the problem. The problem is below:A Coast Guard cutter detects an unidentified ship at a distance of 17.0 km in the direction 15.0° east of north. The ship is traveling at 20.0 km/h on a course at 40.0° east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. (a) If the speedboat travels at 48.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north. (b) Find the time required for the cutter to intercept the ship in minutes.
Explanation / Answer
(a) Set the origin of your coordinate system as the position of the Coast Guard. The position vector of the ship can be mapped with respect to the position of the coast guard as (17cos(75) + 20tcos(50)) i + ((17sin(75)+20tsin(50)) j (NOTE: angles are taken with respect to east because cosines and sines are easier to calculate this way). At the time of interception, the position of the coast guard equals the position of the speedboat at the same moment. Thus, the following component equations hold:
48tcos() = 17cos(75)+20tcos(50)
48tsin() = 17sin(75) + 20tsin(50)
Take first equation and solve for t in terms of theta.
t = (17cos(75))/(48cos()-20cos(50))
Plug this value of t into the second equation. You will get two sides of an equation, both of which look extremely ugly. Go to your calculator and press the Y= button. Then, in Y1, input the first side of your equation. In Y2, input the second side of your equation. Use the intersect feature of your calculator to find the value of that corresponds to the intersection of the two graphs and that hence satisfies the above system of equations. Recall that theta gives you the bearing with respect to east, so you'll have to convert it to north by taking -90 degrees. A negative value implies that your value is that many degrees east of north. A positive value implies that your value is that many degrees west of north.
(b) To obtain the ultimate time value where the ship is intercepted, plug in the value of that you obtained after the intersection calculation in part a into the function of t with respect to theta that you obtained in part a. The value of t you obtain is the answer.