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!