Unidentified Unit
PHYSICS: (URGENT!) TOPIC: RELATIVE MOTION?
HELP: I missed a week of our Physics class due to flu and now, my professor told me that he'll excuse me if I am able to answer this problem. He'll be expecting me to present this problem and the solutions to the whole class by Wednesday.
Problem:
A Coast guard cutter detects an unidentified ship at a distance of 20.0 km in the direction 15.0 degrees east of north. The ship is traveling at 26.0 km/h on a course at 40.0 degrees east of north. The Coast Guard wishes to send a speedboat to intercept the vessel and investigate it. If the speed boat travels 50.0 km/h, in what direction should it head? Express direction as a compass bearing with respect to due north and using unit vector notation.
The unidentified ship starts at a distance of 20 km, at 15º east of north,
with a velocity of 26 km/h, at 40º east of north.
These (r,θ) polar coordinates can be transformed (x,y) coordinates by using the following formulas.
x = r sinθ
y = r cosθ
So the ship unidentified starts at position ( 5.18x + 19.3y ) km
with a velocity of ( 16.7 x + 19.9y) km/h
The speedboat starts at corrdinates (0,0), and has an initial velocity of (50.0 , φ) in polar coordinates. In cartesian, this becomes ( 50sinφ , 50cosφ )..
At a given time t from the start, the x-coordinate of the ship will be 5.18 + 16.7t, while that of the speedboat will be 50sinφ*t.
The y-coordinate of the ship will be 19.3 + 19.9t while that of the speedboat will be 50cosφ*t.
We want the solution for which the equations
5.18 + 16.7t = 50sinφ*t
19.3 + 19.9t = 50cosφ*t
are both satisfied. This is a system of two independdant equations with two unknowns, which can be solved.
Taking the first equation, we have
t = 5.18 / (50sinφ - 16.7)
Plugging it into the second one gets us
19.3 = (50cosφ - 19.9)t
19.3 = (50cosφ - 19.9) * 5.18 / (50sinφ - 16.7)
3.732 = (50cosφ - 19.9) / (50sinφ - 16.7)
3.732 = (cosφ - 0.398) / (sinφ - 0.334)
3.732 * (sinφ - 0.334) = (cosφ - 0.398)
3.732 sinφ - 1.247 = cosφ - 0.398
3.732 sinφ = cosφ + 0.849
Let's take the square of each side
13.93 sinφ^2 = cosφ^2 + 1.70 cosφ + 0.721
Knowing that [sinφ^2 + cosφ^2 = 1], we have
13.93 (1 - cosφ^2) = cosφ^2 + 1.70 cosφ + 0.721
13.93 - 13.93 cosφ^2 = cosφ^2 + 1.70 cosφ + 0.721
14.93 cosφ^2 + 1.70 cosφ - 13.21 = 0
This is a simple quadratic equation in cosφ.
The solutions are
cosφ = 0.885
cosφ = -0.999
acos (0.885) = 27.7º
acos (-0.999) = 177º (clearly not what we're looking for)
So it seems the direction in which to head is about 27.7º east of north
As in the start, this direction can be expresed in (x,y) coordinates by using
x = r sinθ ; y = r cosθ
so it becomes; v = (23.2x + 44.3y) km/h
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