Author Topic: Parametric Eq Q - please help! xD (Read 744 times) Tweet Share

VCE1996 · « **on:** April 14, 2014, 06:12:15 pm »

Hi,

I have an extended response question that I'm not too sure how to do. if someone could please explain and help out that would be fantastic! thanks

The motion of 2 spheres is given by the parametric equations for t E R (t is an element of all real numbers)

r1 (t) = (2sin(2t)-3)i + (cos(2t)-2)j r2 (t) = (t+2)i + (t^2 + 2t)j

1. Find the cartesian equation of the paths described by r1 and r2

2. Sketch the trajectories of the two spheres on the same graph, indicating the direction of increasing t

3. Find the distance between the two spheres when t=0

e^1 · « **Reply #1 on:** April 14, 2014, 07:45:05 pm »

1.

$\begin{alignedat}{1} \vec{r_1} (t) = (2 \sin(2t) -3)\hat{i} +(\cos(2t) - 2)\hat{j} \\\\ \text{Let } x = 2 \sin(2t) - 3 \\ y = \cos(2t) - 2 \\\\ \frac{x+3}{2} &= \sin(2t) \\ \implies \left(\frac{x+3}{2}\right)^2 &= \sin^2 (2t) \\ &= 1 - \cos^2 (2t) \\\\ \implies \cos (2t) = \pm\sqrt{1 - \left(\frac{x+3}{2}\right)^2} \\\\ \therefore y &= \pm\sqrt{1 - \left(\frac{x+3}{2}\right)^2} - 2 \\ &= \pm\sqrt{-\frac{(x+5)(x+1)}{4}} - 2 \end{alignedat}$

$\begin{alignedat}{1} \vec{r_2} (t) = (t+2)\vec{i} + (t^2 +2t)\vec{j} \\\\ \text{Let } x = t +2 \text{ and } y=t^2 + 2t \\\\ x^2 = (t +2)^2 &= t^2 + 4t + 4 \\ &= (t^2 + 2t) + 2t + 4 \\ x^2 &= y + (2t + 4) \\\\ \text{Note that } 2x = 2t + 4 \\\\ \therefore x^2 = y + 2x \\ \implies y = x^2 - 2x \end{alignedat}$

I'll leave the two questions to you. A hint that for the second question, one of them should be familiar to you, while the other Cartesian equation should be easy to sketch because of its domain. For the third question, find r1(0) and r2(0) and use those to find the distance.

kinslayer · « **Reply #2 on:** April 14, 2014, 08:00:38 pm »

Here is an alternative solution for the cartesian equation for $r_1(t)$ :

e: woops, typo

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Author Topic: Parametric Eq Q - please help! xD (Read 744 times) Tweet Share

VCE1996

Parametric Eq Q - please help! xD

e^1

Re: Parametric Eq Q - please help! xD

kinslayer

Re: Parametric Eq Q - please help! xD

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