# Find maximum acceleration given the displacement equation

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1. Apr 30, 2017

### edgarpokemon

1. The problem statement, all variables and given/known data
Given:
x=9sin2t +16cos2t+100

The problem asks me for the maximum acceleration and the maximum velocity.

2. Relevant equations
dx/dt=v
dv/dt=a

3. The attempt at a solution
The problem asks me for the maximum acceleration and the maximum velocity. To find the maximum velocity, i set A equal to 0 and i solve for t. But when i substitute the found t in the acceleration, it does not equal zero as it should. I did got the max velocity right though, but i have no idea how to find the maximum acceleration, i could use mass and newton laws, but no mass is given. just the x function. is a dynamics question. help!!!

Last edited by a moderator: Apr 30, 2017
2. Apr 30, 2017

### Staff: Mentor

Can you show your work in detail?

3. Apr 30, 2017

### edgarpokemon

x=9sin2t +16cos2t + 100
dx/dt= 18cos2t - 32sin2t = v
dv/dt= -36sin2t - 64cos2t = a

when a=0, v=maximum
-36sin2t - 64cos2t =0
2t=arctan (-64/36)
t= either -1.05 or (pi-1.05), i chose the second one which is 2.08

so at t=2.08, the acceleration should be zero, but it isnt, -36sin (2.08) - 64cos (2.08) does not equal zero, "why not?". at t=2.08, the maximum velocity is reached and that is correct.

4. Apr 30, 2017

### Staff: Mentor

Shouldn't that be 2t = ...?

Also, you should keep a few extra digits of precision in intermediate values, particularly when dealing with angles. The trig functions can be pretty sensitive to small changes in argument in certain regions of their domains.

5. Apr 30, 2017

### edgarpokemon

right! thanks!, but how do i calculate the maximum acceleration?

6. Apr 30, 2017

### rcgldr

Calculate da/dt and then solve for da/dt = 0 (could be a minimum or a maximum).

7. Apr 30, 2017

### Staff: Mentor

You could differentiate again (finding what is called the "jerk") and do what you did for finding the maximum velocity.

Another approach that is a tad more subtle is to note that the expressions for position, velocity, and acceleration all have similar form to the well known angle sum and difference identities. Remember them? One such looks like:

$sin(A - B) = sin(A) cos(B) - cos(A) sin(B)$

If you can find a scaling factor for your function that turns the numerical coefficients of the sin(2t) and cos(2t) into the cos and sin of some angle, then the scaling factor will give you the maximum value.

8. Apr 30, 2017

### edgarpokemon

aaa thank you so much!! i like the jerk option better lol

9. Apr 30, 2017

### edgarpokemon

thanks!!

10. Apr 30, 2017

### edgarpokemon

for the second option, i guess you would have to use linear algebra and find an eigenvalue correct?

11. Apr 30, 2017

### Staff: Mentor

No, the derivation uses the trig identity form to find the scaling factor that makes the expression equivalent to that scaling factor multiplied by a sine or cosine function. Since the sine or cosine always has a magnitude of 1 (so that it has extremes of +1 and -1), that means the scaling factor itself is the maximum amplitude of your function.

The derivation is not difficult and the result is very easy to remember. I'd suggest that you perform the derivation yourself at least once. The end result though is simple enough to commit to memory. Given a function that's the sum of a sine and cosine term:

$f(θ) = a~sin(θ) \pm b~cos(θ)$

the amplitude (or maximum value) of the function is simply $\sqrt{a^2 + b^2}$.

12. Apr 30, 2017

### edgarpokemon

thank you!! i really really appreciate it, i like knowing other ways to solve a problem, thank you!!