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Carnegie Mellon University

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Problem 24 Hard Difficulty

A Coast Guard cutter detects an unidentified ship at a distance of 20.0 $\mathrm{km}$ in the direction $15.0^{\circ}$ east of north. The ship is traveling at 26.0 $\mathrm{km} / \mathrm{h}$ on a course at $40.0^{\circ}$ east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. (a) If the speedboat travels at 50.0 $\mathrm{km} / \mathrm{h},$ in what direction should it head? Express the direction $\mathrm{as}$ a compass bearing with respect to due north. (b) Find the time required for the cutter to intercept the ship.

Answer

a. Check Image for solution
b. Check lmage for solution

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Video Transcript

be the diagram of the system itself. We can say that then. Ah, the displacement vector. Someone in the x X component of the first displace in vector is there, Uh, the why component of the first displacement vector is gonna be equaling two dese of one, the magnitude. And so we can say that the displacement vector decent to makes an angle with the Y axis, this angle is 40 degrees minus 15 degrees, equaling 25 degrees. And so we can say that the displacement vectors of two in the ex direction equals decent, too Sign of 25 degrees. Um, we can say decent to Andy. Why would be equal to decent to co sign of 25 degrees? And we can say that decent to itself. The magnitude of the second displacement vector is visa to the speed of the ship multiplied by t. And so we can say that then decent three in the extraction, this would be decent. Three sign of Alfa Alfa is that angle that you see here and then we have the third displacement vector in the wind direction. This would be T's of three co sign of Alfa knowing that decent three is equaling the speed of the speedboat Visa. Three times teeth. And so we can then say that here, uh, dese of three x equals simply d someone x plus D. So two ex And we can then say that this is gonna be zero and that here Piece of three, sign of Alfa It's gonna equal decent to sign of 25 degrees. And so we could say that Alfa is equaling arc sine of visa to t divided by Ivy's of three t. Of course, the teas they're gonna cancel out this would be multiplied by sine of 25 degrees. So this is equaling Arc sine, huh? 26 26.0 kilometers per second, divided by 50.0 kilometers per second time sign of 25 degrees and weakens. Find that Alfa is equaling 12.7 degrees. And so Fada is your equal Alfa plus 15. So 27 0.7 degrees. So this would be our Alfa and are Seita angles. Um, so we could say that the direction of the speedboat, um, is 27.7 degrees east of north and this would be the direction of speedboat to intercept shit. So this would be our answer. This will be your final answer. Four part eh for part B. Then we can do the exact same thing in the UAE direction where we have decent three co sign of Alfa equaling dese of one plus decent too co sign of 25 degrees and we could rearrange for and say that Visa three t co sign of Alfa equals d someone plus the sub to thi co sign of 25 degrees and then re arrange for tea So t is going to be equaling Piece of one divided by Visa three co sign of Alfa minus visa too co sign of 25 degrees. And so the time T is gonna be equaling 20.0 kilometers divided by 50 0.0 kilometers per hour multiplied by co sign of 12.7 degrees. This would be minus 26 0.0 kilometers per hour, both supplied by co sign of 25 degrees. We're going to multiply this by 60 minutes for every one hour and we find that the time required for the speed boat to intercept the ship is 47.6 minutes so this would be our time for part B. That is the end of the solution. Thank you for watching.

Carnegie Mellon University
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