Intersecting and Colliding Curves

Oct 2012
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USA
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The particles don't collide because no value of t satisfies all three sets of parametric equations set equal to each other. Would the next step of seeing which points intersect involve solving the parametric equations for t?
 
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romsek

MHF Helper
Nov 2013
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California
what you have to realize is that the particles might not travel at the same rate.

So you can't just equate the components.

What you can do is let

$r_1(t)=<t, t^2, t^3>$

and

$r_2(s) = < 1+2s, 1+6s, 1+14s>$

and see if the 3 equations (each component) in two unknowns, $s$ and $t$, have a solution.

As it turns out there are 2 solutions, or points where the space curves intersect.

Colliding on the other hand requires that the two particles' paths intersect at a common time.

In this case you would just equate the components and as you saw there was no collision solution.

See if you can solve for the intersections.
 
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Oct 2012
1,314
21
USA
t = an s equation. That s equation is then plugged into the t^{2} equation. Next the equation is solved for s. The s values are plugged back into the s equations to get the points which are 1,1,1 and 2,4,8.
 
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