Show that he object moves on an elliptical path

[Saladsamurai]

Okay then

Homework Statement

An object moves in the xy-plane such that its position vector is

$$\bold{r} = \bold{i}a\cos(\omega t)+\bold{j}b\sin(\omega t) \qquad (1)$$

where a,b, and $\omega$ are constants.

Show that the object moves on the elliptical path

$$(\frac{x}{a})^2+(\frac{y}{b})^2 =1 \qquad (2)$$


I have never studied ellipses, so I am 'googling' them now as we speak. I can see that (2) resembles the equation of a circle except that it includes a couple of scaling factors 'a' and 'b'.

I am just not sure how to relate (1) and (2) to each other.

Can I get a friendly 'nudge' here?

Thanks!

~Casey

 

[Feldoh]

$$\bold{r} = \bold{i}a\cos(\omega t)+\bold{j}b\sin(\omega t) = \bold{i}x +\bold{j}y$$

Right?

So $$x = a\cos(\omega t), y = b\sin(\omega t)$$

Just plug it into the equation (2) in order for it to hold it must hold for all t.

 

[Saladsamurai]

$$\bold{r} = \bold{i}a\cos(\omega t)+\bold{j}b\sin(\omega t) = \bold{i}x +\bold{j}y$$

Right?

So $$x = a\cos(\omega t), y = b\sin(\omega t)$$

Just plug it into the equation (2) in order for it to hold it must hold for all t.

Right. I thought I was making it more difficult than it is. I don't know why I thought that there was something more to it. Thanks again!