Question on finding maximum magnitude of acceleration

[RoboNerd]

Homework Statement

An object is moving in the xy-plane according to the equations x(t) = 3sin(3t) and y(t) = 4cos(3t). What is the maximum magnitude of the particle's acceleration?

1. 1) 5 m/s2
2. 2) 15 m/s2
3. 3) 30 m/s2
4. 4) 36 m/s2 [the accepted answer]
5. 5) 45 m/s2
   

Homework Equations

x(t) = Asin(wt + phase constant)

The Attempt at a Solution

So, I know that for each dimension if I take the second derivative of the two position equations I get acceleration, and the maximum acceleration for each of those two is simply A*omega^2.

So that's what I did.

For acceleration in the x-dimension, I get: 3*3^2.
For acceleration in the y-dimension, I get 4*3^2.

Taking the squares of the two acceleration components and then summing them and taking the square root of the sum gives 45, which is what I got. Why do the authors then believe 36 m/sec^2 is the right answer?

Thanks in advance for the input!

 

[TSny]

Does a_{x, max} occur at the same instant that a_{y, max} occurs?

 

[RoboNerd]

no, of course not. how would I fix for that, then?

 

[RoboNerd]

But how would I be able to tell if the aymax and axmax occur simultaneously?

 

[TSny]

Can you find an expression for the magnitude of the (total) acceleration as a function of time?

 

[RoboNerd]

Yes, take the squares of the two equations, sum and square root.

I am too lazy to write it out here, unless it is really needed. ;-)

 

[TSny]

But how would I be able to tell if the aymax and axmax occur simultaneously?

Find expressions for a_x and a_y as functions of time. By inspection you will see that their max values do not occur at the same time.

 

[drvrm]

For acceleration in the x-dimension, I get: 3*3^2.
For acceleration in the y-dimension, I get 4*3^2.

are your accelerations correctly written-
i think there are cosine /sine terms also in the accelerations.
check.
or you can find r the radius vector and find out the acceleration and find out the maximum.

 

[RoboNerd]

Aha, OK. thanks.

Find expressions for ax and ay as functions of time. By inspection you will see that their max values do not occur at the same time.

 

[RoboNerd]

are your accelerations correctly written-
i think there are cosine /sine terms also in the accelerations.
check.
or you can find r the radius vector and find out the acceleration and find out the maximum.

I was just doing manipulations of maximum and minimum values [just the amplitudes of the two equations... if you know what I mean]

 

[TSny]

Yes, take the squares of the two equations, sum and square root.

I am too lazy to write it out here, unless it is really needed. ;-)

That's pretty lazy.
I wouldn't have suggested it unless I thought it was worth the trouble.

 

[RoboNerd]

sqrt( 3sin(3t)^2 + 4cos(3t)^2 )

Here you go, sir!

 

[TSny]

No, that's not the sum of the squares of the acceleration components.
$a = \sqrt{a_x^2+a_y^2}$