A ship leaves the island of Guam and sails a distance 295km at an angle 35.0 north of west.
a:In which direction must it now head so that its resultant displacement will be 100km directly east of Guam? (Express your answer as an angle measured south of east)
b:How far must it sail so that its resultant displacement will be 100km directly east of Guam?
If it started at O (0,0), then after sailing NW, it arrives at P(-295 cos 35°, 295 sin 35°)
Now we want to arrive at point Q= (100,0)
SO, use the law of cosines to get the distance PQ: PQ² = 295² + 100² - 295*100*cos(180-35)
Now, use the law of sines to get the angle OPQ:
sin 145°/PQ = sin OPQ/100
Now, the heading is 55° + OPQ° east of south
To find the direction the ship must now head, we need to determine the angle south of east.
a: To find the angle south of east, we can subtract the given angle of 35.0° north of west from 180° (since east is 180°).
180° - 35.0° = 145.0°
Therefore, the ship must now head at an angle of 145.0° south of east.
b: To find out how far the ship needs to sail for the resultant displacement to be 100km directly east of Guam, we need to use vector addition.
Note that the displacement to the west (295km at an angle of 35.0° north of west) can be represented as the vector (-295, -35.0°).
Let the distance the ship needs to sail be represented by the magnitude "d" and angle θ. The displacement of this leg can be written as the vector (d, θ).
The resultant displacement can be calculated by adding the two vectors:
(-295, -35.0°) + (d, θ) = (100, 0°)
To simplify the calculation, we can break down the vectors into their x and y components:
(-295 cos(35.0°), -295 sin(35.0°)) + (d cos(θ), d sin(θ)) = (100, 0)
This gives us two equations:
-d cos(35.0°) + d cos(θ) = 100
-d sin(35.0°) + d sin(θ) = 0
By solving these equations simultaneously, we can find the value of d and θ.
To solve this problem, we can use vector addition. Let's break down the information given:
a) The ship sailed a distance of 295 km at an angle of 35.0° north of west. We can represent this displacement as a vector by breaking it down into its horizontal and vertical components.
Horizontal component = 295 km * cos(35.0°)
Vertical component = 295 km * sin(35.0°)
Now, we need to find the remaining displacement (RD) required to end up 100 km directly east of Guam. Since it needs to be directly east, the y-component of RD will be 0. The x-component of RD will be 100 km.
Now, let's calculate the horizontal and vertical components of RD:
Horizontal component of RD = 100 km * cos(0°) = 100 km
Vertical component of RD = 100 km * sin(0°) = 0
The horizontal component of the resultant displacement (RD) should be equal to the sum of the horizontal components of the initial displacement and RD. Similarly, the vertical component of RD should be equal to the sum of the vertical components of the initial displacement and RD.
Hence, we can write the following equations:
Horizontal component of the resultant = Horizontal component of the initial displacement + Horizontal component of RD
Vertical component of the resultant = Vertical component of the initial displacement + Vertical component of RD
Let's plug in the values we have:
Horizontal component of the resultant = 295 km * cos(35.0°) + 100 km
Vertical component of the resultant = 295 km * sin(35.0°) + 0
Now, we can find the magnitude and direction of the resultant displacement.
Magnitude of the resultant displacement = sqrt((Horizontal component of the resultant)^2 + (Vertical component of the resultant)^2)
Direction of the resultant displacement = arctan(Vertical component of the resultant / Horizontal component of the resultant)
b) In this case, we need to find the magnitude of RD. We can use the same equations as before:
Horizontal component of the resultant = 295 km * cos(35.0°) + RD
Vertical component of the resultant = 295 km * sin(35.0°) + 0
Now, we can calculate the magnitude of the resultant displacement:
Magnitude of the resultant displacement = sqrt((Horizontal component of the resultant)^2 + (Vertical component of the resultant)^2)
To summarize:
a) The ship must head in a direction measured south of east, and the magnitude of the resultant displacement will be the magnitude calculated using the above equations.
b) The ship needs to sail a distance equal to the magnitude of the resultant displacement calculated using the above equations.