the Position Vector

[nasi112]

The coordinates of an object moving in the xy plane vary with time according to the equations [MATH]x(t) = 6t^2 - 12t[/MATH] and [MATH]y(t) = t^3 - 9t[/MATH], where [MATH]x[/MATH] and [MATH]y[/MATH] are in meters, and [MATH]t[/MATH] is in seconds. The position vector [MATH]\vec{r}[/MATH] of the object when [MATH]a_x = 2a_y[/MATH] is:

(a) [MATH]-6i - 8j[/MATH]
(b) [MATH]15i + 18j[/MATH]
(c) [MATH]48i + 28j[/MATH]
(d) [MATH]-3.3i - 2.9j[/MATH]
(e) [MATH]-4.5i - 4.4j[/MATH]

I tried to solve it in this way
[MATH]\vec{r} = x(t) + y(t) = 2y(t) + y(t) = (2t^3 - 18t)i + (t^3 - 9t)j[/MATH]
[MATH]x(t) = 2y(t)[/MATH]
[MATH]6t^2 - 12t = 2(t^3 - 9t)[/MATH]
[MATH]3t^2 - 6t = t^3 - 9t[/MATH]
[MATH]0 = t^3 - 3t^2 - 3t[/MATH]
[MATH]t = \frac{3 + \sqrt{21}}{2} \ [/math] or [math] \ t = \frac{3 - \sqrt{21}}{2} \ [/math] or [math] \ t = 0 \ [/MATH]
None of these will give the answer in the multiple choices! Did I do anything wrong?

 

[Steven G]

Is that all the work you did? You found a value for t but they ask for r(t).

Can you explain why you got that x(t) = 2y(t)?

 

[skeeter]

the problem statement says “when [MATH]a_x = 2a_y[/MATH], not when [MATH]x(t) = 2y(t)[/MATH]
[MATH]a_x(t) = x’’(t)[/MATH]
[MATH]a_y(t) = y’’(t)[/MATH]
try again ...

 

[nasi112]

Opps, I forgot that [MATH]a_x[/MATH] and [MATH]a_y[/MATH] are the components of the acceleration

I got it now. Thank you so much.

 

[HallsofIvy]

Saying that \(\displaystyle a_x= 2a_y\) does NOT mean that \(\displaystyle x= 2y\) if that is where you got "\(\displaystyle x(t)= 2y(t)\)" if that was where you got that! \(\displaystyle a_x\) and \(\displaystyle a_y\) are the components of the acceleration vector, not the position vector.

Saying that the position vector is given by "according to the equations \(\displaystyle x(t)=6t^2−12t\) and \(\displaystyle y(t)=t^3−9t\)" means that the position vector is \(\displaystyle (6t^2- 12t)\vec{i}+ (t^3- 9t)\vec{j}\). The velocity vector is \(\displaystyle (12t- 12)\vec{i}+ (3t^2- 9)\vec{j}\) and the acceleration vector is \(\displaystyle 12\vec{i}+ 6t\vec{j}\).

So saying that \(\displaystyle a_x= 2a_y\) means that \(\displaystyle 12= 2(6t)= 12t\).