Question

A particle starts at the origin and moves along the curve y=2x3/2 in the positive x-direction at a speed of 3 cm/sec, where x,y are in cm.
find the position of the particle at t=6.
Now the answers are x = 8.221 and y = 15.714 but how you get there remains a mystery to me. Thanks for helping

Answer 1

Slight edit it's y = (2/3)*x^(3/2)

Answer 2

Since it's talking about three variables: x, y, and t. Perhaps parametric equations?

Answer 3

Hmm, hold on.. does it mean the speed along the curve is 3cm/s or just the speed in the X direction is 3cm/s?

Answer 4

To be honest I'm not quite sure, these questions are very poorly written and this is exactly how it is in the book

Answer 5

I tried it with the assumption that the speed was in the X direction only and didn't get those numbers you listed, so I guess it must be the other way.
(that complicates things . . .)

Answer 6

I've been working on this one forever and I just don't know how to get to those results

Answer 7

You know what I mean by parametric functions, though, right?
Need to find x=f(t) and y=g(t)=(2/3)f(t)^(3/2).

Answer 8

oh yeah of course

Answer 9

If it's speed along the curve, then makes me think that finding arc length will be necessary.

Answer 10

well arc length would be integral from a to b for speed

Answer 11

and speed is sqrt((x')^2 + (y')^2) or sqrt(1+ t) or at least that's what i'm getting

Answer 12

If that x' and y' are derivatives with respect to t, then yes, I think that's correct, but we don't have x and y as functions of t yet.

This sounds like a big pain in the neck, but I think this strategy might work:
\[arc \space length, \space s=\int\limits_{0}^{x} \sqrt{1+(\frac{dy}{dx})^2} dx\]
Distance = rate × time, and the rate is 3cm/s for 6s, so s=18cm.
Set that integral equal to 18 and solve for the upper limit, x.

Answer 13

well we can just say x = t and y = (2/3)t^(3/2)

Answer 14

I'm going to toss up the Bat-Signal.
@TuringTest @ash2326 @experimentX @hartnn
Second opinion on this, guys?

Answer 15

We don't know if x=t is the proper function. I tried x=3t when I assumed that the speed was only in the X direction at first, but that didn't work.

Answer 16

If x=t then the position at t=6 would be (6, 9.797..)

Answer 17

The arc length formula isn't too bad here because (dy/dx)^2 = x. The integral isn't very messy.

Answer 18

Dang, I think I'm missing something. I tried it and got exactly 8 for x...
So close . . .