The motion of a spring that is subject to dampening (such as a car's shock absorber)is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion for a point on such a spring is
s(t)=3∗e−2tsin(3ðt)
where t is given in seconds.
a. Find the velocity of the point after t seconds.
I found the derivative for a to be e^(-2t)(9picos(3pit)-6sin(3pit))
for part b I have to find t when the velocity=0. I keep getting .2122 on my ti 89 but I know that's not the right answer. Can someone at least show me by hand how to solve?
s(t) = 3e^-2t sin 3πt
s'(t) = e^-2t (9π cos 3πt - 6sin 3πt)
so far, so good
when s'(t) = 0, we have
3π cos 3πt - 2sin 3πt = 0
t = 0.1445
If you visit
http://rechneronline.de/function-graphs/
and enter
exp(-2x) * sin(3*pi*x)
for your function, you can see that's where it changes direction. You can also plot the derivative, to verify that.
wolframalpha.com gives the solution as
0.21221(nπ + 0.68085)
Better see whether your TI calculator is also tacking on some extra factors.
To find the velocity of the point after t seconds, we need to find the derivative of the function s(t). As you mentioned, you have already found the derivative, which is:
s'(t) = e^(-2t) * (9πcos(3πt) - 6sin(3πt))
Now, let's move on to part b where we need to find the value of t when the velocity equals zero (s'(t) = 0).
To solve this equation, we'll set s'(t) equal to zero and solve for t:
e^(-2t) * (9πcos(3πt) - 6sin(3πt)) = 0
Now, there are two cases to consider:
Case 1: e^(-2t) = 0
For this case, since e^(-2t) is always positive and never zero, there are no solutions.
Case 2: 9πcos(3πt) - 6sin(3πt) = 0
We can simplify this equation by dividing both sides by 3π:
3cos(3πt) - 2sin(3πt) = 0
Now, let's solve this equation:
Divide both sides by sqrt(3^2 + (-2)^2) = sqrt(9 + 4) = √13:
(3/√13)cos(3πt) - (2/√13)sin(3πt) = 0
Using the trigonometric identity sin(x - π/6) = cos(x)sin(π/6) - sin(x)cos(π/6), we can rewrite the equation as:
sin(3πt - π/6) = 0
To find the values of t, we need to find the solutions for sin(3πt - π/6) = 0, which occurs when the argument of the sine function is an integer multiple of π:
3πt - π/6 = nπ, where n is an integer
Solving for t:
3πt = nπ + π/6
t = (n + 1/18) / 3
So, the values of t when the velocity is zero are given by:
t = (n + 1/18) / 3, where n is an integer.
Please note that the numerical value you obtained, 0.2122, might be correct depending on the value of n.