Find maximum acceleration given the displacement equation

[edgarpokemon]

Homework Statement

Given:
x=9sin2t +16cos2t+100

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

Homework Equations

dx/dt=v
dv/dt=a

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!

 

[gneill]

Can you show your work in detail?

 

[edgarpokemon]

Can you show your work in detail?

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.

 

[gneill]

2t=arctan (-64/36)
t= either -1.05 or (pi-1.05), i chose the second one which is 2.08

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.

 

[edgarpokemon]

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.

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

 

[rcgldr]

how do i calculate the maximum acceleration?

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

 

[gneill]

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

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.

 

[edgarpokemon]

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.

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

 

[edgarpokemon]

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

thanks!

 

[edgarpokemon]

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.

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

 

[gneill]

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

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 $\sqrt{a^2 + b^2}$.

 

[edgarpokemon]

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 $\sqrt{a^2 + b^2}$.

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