Suppose that the position functions of two particles, P1...

Suppose that the position functions of two particles, $P_{1}$ and $P_{2},$ in motion along the same line are $$s_{1}=\frac{1}{2} t^{2}-t+3 \quad \text { and } \quad s_{2}=-\frac{1}{4} t^{2}+t+1$$ respectively, for $t \geq 0$ (a) Prove that $P_{1}$ and $P_{2}$ do not collide. (b) How close do $P_{1}$ and $P_{2}$ get to each other? (c) During what intervals of time are they moving in opposite directions?

Transcript

00:01 So in this question we have s1, which is a half t squared minus t plus 3.

00:09 And we have s2, which is minus a quarter t squared plus t plus 1.

00:16 And we want to first out, first off show that they don't collide.

00:21 So we've got t greater than or equal to zero for both.

00:25 So a collision happens if s1 equals s2.

00:32 So that would be a half t squared minus t plus three equals minus a quarter t squared plus t plus one.

00:43 So now putting everything over onto this side, we get three quarters t squared minus two t plus two equals zero.

00:57 And we want to show that this has no positive solutions.

01:01 So let's find the solutions.

01:03 Let's multiply by four thirds.

01:06 Let's multiply by two thirds to get a half t.

01:08 Squared minus four thirds t plus four thirds equals zero.

01:16 Then this is solved by t equals four thirds plus or minus the square root of 16 over nine minus four ac.

01:29 So that's minus two times four over three.

01:38 So this is four thirds plus or minus root 16 over nine minus eight.

01:44 Over 3, but 8 over 3 is bigger than 16 over 9, which means that this square root is less than 0, so we have no real solutions, and therefore no collision.

02:10 So that's that.

02:12 Now how close do they get? well, the separation, delta s, is s1 minus s2, which is three quarters t squared, minus 2 t plus 2.

02:26 And how close do they get is the question, what is the minimum value of delta s? so let's find out when this is minimal...

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