A ship, proceeding southward on a straight course at the rate of 12 miles/hr is, at noon,
40 miles due north of a second ship, which is sailing west at 15 miles/hr.
a) How fast are the ships approaching each other 1 hour later?
b) Are the ships approaching each other or are they receding from each other at 2o’clock and at what rate?
I see 4 questions posted in quick succession. You have given no indication of any start to these on your part, nor have you told us where your problems are.
This looks like an assignment or homework.
Surely you realize that if just do these questions for you, you will not learn anything.
How did you start these ?
To answer these questions, we need to understand the concept of relative motion. We can consider the two ships as individual reference points and analyze their displacement with respect to each other.
First, let's define the situation at noon. Ship A is 40 miles due north of Ship B, and Ship B is sailing west. Therefore, the initial distance between the two ships is the hypotenuse of a right-angled triangle formed by their positions.
Using the Pythagorean theorem, we can calculate the initial distance as follows:
Initial distance = √((40^2) + (0^2))
= √1600
= 40 miles
a) To determine how fast the ships are approaching each other 1 hour later, we need to find the change in their distance over time. Let's consider Ship A first.
Ship A is moving southward at the rate of 12 miles/hr. Hence, after 1 hour, Ship A will be 12 miles south of its initial position. This means its new position will be (0, -12).
Now, let's consider Ship B. Ship B is moving westward at the rate of 15 miles/hr. Therefore, 1 hour later, Ship B will be 15 miles further west of its initial position. This means its new position will be (-15, 0).
We can now calculate the distance between the two ships 1 hour later. Again, using the Pythagorean theorem:
Distance after 1 hour = √((0 - (-15))^2 + ((-12) - 0)^2)
= √((15)^2 + (12)^2)
= √(225 + 144)
= √369
≈ 19.21 miles
The initial distance between the ships was 40 miles, and after 1 hour, it becomes approximately 19.21 miles. Therefore, the change in distance is:
Change in distance = Initial distance - Distance after 1 hour
= 40 - 19.21
≈ 20.79 miles
Since the change in distance is positive, it means the ships are approaching each other.
b) To determine if the ships are approaching or receding from each other at 2 o'clock, we need to analyze their positions and velocities.
At 2 o'clock, Ship A will be 24 miles south of its initial position (12 miles/hr x 2 hr = 24 miles). Therefore, its new position will be (0, -24).
At the same time, Ship B will be 30 miles west of its initial position (15 miles/hr x 2 hr = 30 miles). Therefore, its new position will be (-30, 0).
Now, let's calculate the new distance between the ships at 2 o'clock:
Distance at 2 o'clock = √((0 - (-30))^2 + ((-24) - 0)^2)
= √((30)^2 + (24)^2)
= √(900 + 576)
= √1476
≈ 38.39 miles
The initial distance between the ships was 40 miles, and at 2 o'clock, it becomes approximately 38.39 miles. Therefore, the change in distance is:
Change in distance = Initial distance - Distance at 2 o'clock
= 40 - 38.39
≈ 1.61 miles
Since the change in distance is positive but very small, it means the ships are still approaching each other but at a much slower rate compared to before.
In conclusion:
a) The ships are approaching each other at a rate of approximately 20.79 miles/hr after 1 hour.
b) At 2 o'clock, the ships are still approaching each other but at a much slower rate of approximately 1.61 miles/hr.