Question on finding maximum magnitude of acceleration

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1. May 8, 2016

RoboNerd

1. The problem statement, all variables and given/known data
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

2. Relevant equations
x(t) = Asin(wt + phase constant)

3. 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!

2. May 8, 2016

TSny

Does ax, max occur at the same instant that ay, max occurs?

3. May 8, 2016

RoboNerd

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

4. May 8, 2016

RoboNerd

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

5. May 8, 2016

TSny

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

6. May 8, 2016

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. ;-)

7. May 8, 2016

TSny

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.

8. May 8, 2016

drvrm

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.

9. May 8, 2016

RoboNerd

Aha, OK. thanks.

10. May 8, 2016

RoboNerd

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

11. May 8, 2016

TSny

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

12. May 8, 2016

RoboNerd

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

Here you go, sir!

13. May 8, 2016

TSny

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