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^(−2t)sin(3ðt)
where t is given in seconds.
a. Find the velocity of the point after t seconds.
v = .
b. Graph the velocity function and find the first time the velocity is 0.
t =
for velocity, just take the derivative:
v(t) = -3e^(-2t) (2sin 3πt - 3πcos 3πt)
To find the velocity of the point after t seconds, we need to take the derivative of the displacement equation with respect to time. Differentiating the equation s(t) = 3e^(-2t)sin(3πt) using the product rule, we get:
s'(t) = 3e^(-2t) * [(-2)sin(3πt) + 3πcos(3πt)]
Now, simplifying this expression further:
s'(t) = -6e^(-2t)sin(3πt) + 9πe^(-2t)cos(3πt)
Therefore, the velocity function is:
v(t) = -6e^(-2t)sin(3πt) + 9πe^(-2t)cos(3πt)
To graph the velocity function and find the first time when the velocity is 0, we can use a graphing calculator or software. Here, we'll use a Desmos online graphing calculator:
1. Open the Desmos graphing calculator (https://www.desmos.com/calculator).
2. Enter the velocity function: -6e^(-2t)sin(3πt) + 9πe^(-2t)cos(3πt)
3. Specify the domain range for t, let's say from 0 to 10 seconds.
4. Adjust the zoom and view settings to clearly visualize the graph.
5. Observe the graph to find the first time the velocity is 0 – this corresponds to the x-intercept or the point(s) where the graph crosses the x-axis.
Note: The velocity function is a combination of exponential and trigonometric functions, so solving for t analytically may not provide an exact solution. Hence, using a graphing calculator or software is more convenient to find an approximate value for t at which the velocity is 0.