How Fast is the Distance Between Two Ships Changing at 4:00 P.M.?

[madah12]

Homework Statement

At noon, ship A is 150 km west of ship B. Ship A is sailing east
at 35 km
h and ship B is sailing north at 25 km
h. How fast is
the distance between the ships changing at 4:00 P.M.?

Homework Equations

The Attempt at a Solution

I did it using geometry and related rates and got 21.5 which was the answer
but I when I try to do it using relative velocity it doesn't work:
I am taking the first ship initial location as the origin
r_ai=0
r_bi=150i
r_a=0+35ti
r_b=150i+25tj
ra/b=a/g +ag/b
ra/b=0+35ti - 150i+25tj
ra/b=(35t-150)i + 25tj
dr(a/b) /dt = 35i+25j
and the magnitude is the way more than 21.5
I feel like I am doing something really wrong or setup the problem wrong.

 

[tiny-tim]

hi madah12!

suppose that a is stationary at (0,0), and b is at cosθti + sinθtj …

then using your method dr/dt = sinθti - cosθtj,

which is not the same as d|r|/dt, is it?

 

[madah12]

oh yes I see d|r|/dt = 0 right cause |r|= 1 for all t? but then does the relative motion law holds for distances if not then I can't solve this problem using relative motion?
Edit
but we can still say that |s|= dD/dt = d ((25t)^2 + (150-35t)^2)^1/2 /dt right?

 

[tiny-tim]

but we can still say that |s|= dD/dt = d ((25t)^2 + (150-35t)^2)^1/2 /dt right?

(have a square-root: √ and try using the X² tag just above the Reply box )

right!

 

[rwisz]

I would suggest checking your calculations and equations again to make sure they are correct. It is possible that you made a mistake in your calculations or set up the problem incorrectly, which is why you are getting a different answer than expected. It is also important to double check the units and make sure they are consistent throughout the problem. Additionally, you may want to try approaching the problem using different methods to see if you get the same answer. If you are still unable to find the correct solution, it may be helpful to seek assistance from a colleague or instructor.