A ship leaves its homeport and sails on a bearing of N28degrees10'E. Another ship leaves the same port at the same time and sails on a bearing of S61degrees50'E. If the first ship sails at 24.0 mph and second sails at 28mph, find the distance between the two ships after 4hrs.
Unless otherwise indicated, all angles are measured CCW from the +x-axis.
d1 = 24mi/h[61.83o]*4h = 96 mi[61.83o]
d2=28mi/h[331.83o]*4h=112 mi[331.83o]
D=d2-d1 = 112mi[331.83o] - 96mi[61.83] =
98.7-52.9i - (45.3+84.6i)=53.4 - 137.5i
= 147.5mi[68.8o] S. of E.
To find the distance between the two ships after 4 hours, we can use the formula:
Distance = Speed × Time
Let's start by calculating the distance traveled by the first ship:
Distance of the first ship = 24.0 mph × 4 hours
Next, let's calculate the distance traveled by the second ship:
Distance of the second ship = 28 mph × 4 hours
Now, we need to find the difference in the coordinates for the ships' destinations.
The first ship sailed on a bearing of N28degrees10'E, which means it traveled 28 degrees 10 minutes east from the north. To convert this to degrees, we can use the fact that there are 60 minutes in a degree. Thus, N28degrees10'E translates to an eastward movement of (28 + 10/60) degrees.
The second ship sailed on a bearing of S61degrees50'E, which means it traveled 61 degrees 50 minutes east from the south. Following the same conversion logic, this translates to (61 + 50/60) degrees eastward.
Now, using these eastward distances, we can calculate the horizontal distance between the two ships by using the cosine rule. The formula for the displacement along the longitude can be formulated as:
(x1 - x2) = (cos θ × distance of the first ship) - (cos φ × distance of the second ship)
where θ is the angle N28degrees10'E and φ is the angle S61degrees50'E.
Now we have the horizontal displacement, but we need to find the vertical displacement. The formula for the displacement along the latitude can be formulated as:
(y1 - y2) = (sin θ × distance of the first ship) + (sin φ × distance of the second ship)
Finally, we can use the Pythagorean theorem to calculate the distance between the two ships:
Distance = √[(x1 - x2)² + (y1 - y2)²]
By plugging in the values we calculated, we can solve for the distance between the two ships after 4 hours.