Two ships leave a harbor at the same time. One ship travels on a bearing s11w at 16 miles per hour. The other ship travels on a bearing n15e at 12 miles per hour. How far apart will the ships be after 2 hours?
I suggest you do this in x y coordinates, x is East and y is North
16 knots for 2 hours is 32 nautical miles
a heading (not a bearing by the way) of South 11 West is 11 degrees clockwise beyond the -y axis
so x location = -32 sin 11 = -6.10
and
y location = - 32 cos 11 = - 31.4
so first ship is at (-6.10 , -31.4 )
The second ship travels N15 E at 12 knots for 2 hours
that is 90 - 15 = 75 deg above x axis
distance = 24 nautical miles
x = 24 cos 75 = 6.21
y = 24 sin 75 = 23.2
so location is (6.21 , 23.2)
now distance between
d^2 = x difference^2 + y difference^2
=
d ^2 = (6.21 + 6.1)^2 + (23.2 + 31.4)^2
I think you can find d now.
the angle between the two courses is 176°
The distance d is thus
d^2 = 32^2 + 24^2 - 2*32*24 cos176°
a very good approximation is just 32+24 = 56 miles, since the two ships are traveling in almost opposite directions.
To find the distance between the two ships after 2 hours, we can use the Pythagorean theorem.
Step 1: Determine the distance each ship has traveled after 2 hours.
Since Ship A travels at a speed of 16 miles per hour, after 2 hours it would have traveled 16 miles per hour * 2 hours = 32 miles.
Since Ship B travels at a speed of 12 miles per hour, after 2 hours it would have traveled 12 miles per hour * 2 hours = 24 miles.
Step 2: Break down the bearing directions into north-south (latitude) and east-west (longitude) components.
Ship A is traveling on a bearing of s11w, which means it is moving south-southeast. The south component is 11 units and the west component is 1 unit.
Ship B is traveling on a bearing of n15e, which means it is moving north-northeast. The north component is 15 units and the east component is 1 unit.
Step 3: Calculate the latitude and longitude distances between the two ships.
The latitude distance is the absolute difference between the north and south components:
Latitude distance = |15 - 11| = 4 units.
The longitude distance is the absolute difference between the east and west components:
Longitude distance = |1 - (-1)| = 2 units.
Step 4: Use the Pythagorean theorem to find the distance between the two ships.
The distance between the two ships is given by:
Distance = √(latitude distance^2 + longitude distance^2)
Distance = √(4^2 + 2^2)
Distance = √(16 + 4)
Distance = √20
Distance ≈ 4.47 units.
Therefore, after 2 hours, the two ships will be approximately 4.47 units apart.