- #1
Adgorn
- 130
- 18
Homework Statement
Hi, everyone. I came across a basic calculus problem concerning movement of 2 problems, I've attempted to solve it using vector analysis and got 1 answer, and then solved it with differentiation and got a different answer. I'll show a version of it I made which is a bit more simplified to make things easier since even then the same problem occurs:
"Particle A moves along the positive horizontal axis, and Particle B moves along the positive vertical axis. At a certain time, the A is at the point ##(5,0)## and moving with speed 3 units/sec; and B is at the point ##(0,3)## and moving with speed 4 units/sec. At what rate is the distance between A and B changing?"
Homework Equations
Pythagorean theorem, chain rule.
The Attempt at a Solution
Intuitively, this is just a "riverboat problem", the rate of change of the distance between A and B at a certain moment is just their speed relative to each other at that moment, and does not depend on their location. To get this relative speed you just have to add the velocity vectors. I have the feeling that this intuition is the fault in my calculation, although I don't realist why, which would explain why the first method produces a wrong answer:
Method 1, vectors: The particles move away from each other with a horizontal speed of 3 units/sec and a vertical speed of 4 units/sec. Thus, the speed at which they move away from each other is ##\sqrt {3^2+4^2}=5## units/sec.
Method 2, differentiation: Let the position of A be ##(a(t),0)##, the position of B be ##(0,b(t))##, the distance between A and B be ##d(t)## and the specified time of the problem be ##t_0##. Then ##a(t_0)=5##, ##a'(t_0)=3##, ##b(t_0)=3## and ##b'(t_0)=4.##
At any moment ##t##, ##d(t)^2=a(t)^2+b(t)^2##, so ##d(t_0)=\sqrt {25+9}=\sqrt {34}##. Differentiating the 2 sides of the equation yields
##2d(t)d'(t)=2a(t)a'(t)+2b(t)b'(t)## or ##d'(t)=\frac {2a(t)a'(t)+2b(t)b'(t)} {2d(t)}##, inserting the values at ##t_0## yields ##d'(t)=\frac {30+24} {2\sqrt {34}}=\frac {27} {\sqrt {34}}##.
The book uses the 2nd method, so I assume something is wrong with the 1st method, probably something very basic that I looked over. The equations of the 2nd method seem to indicate that the relative velocity does depend on the location of the particles, which is something I don't quite understand. Clarification would be very appreciated.