Question

Modeling Exercise 6.4.2 Use the method of undetermined coefficients to find a particular solution y

p
​
(t) to (4.119); this solution will depend on k. Then write out a general solution to (4.119). The general solution should also depend on k. Modeling Exercise 6.4.3 As noted in Example 4.20, a rider who rides off a 1.5 meter drop will hit the ground at about 5.42 meters per second. Solve (4.119) with initial conditions y(0)=0, y
′
(0)=−5.42, to find the displacement y(t) of the shock. This is a function of t that also involves the indeterminate k. Modeling Exercise 6.4.4 Determine, either graphically or analytically, the smallest value k=k
∗
for k that results in the shock compressing no more than −0.14 meters. What is the corresponding value for c
∗
? Modeling Exercise 6.4.5 With the values for k
∗
and c
∗
found in Modeling Exercise 6.4.4, plot the solution y(t) on the interval 0≤t≤1. Also plot y
′′
(t) on this same time interval. What is the largest acceleration to which the rider is subjected? (It may seem large, but we haven't accounted for the shock absorption provided by the tires or the fact that the rider's legs may also act as shock absorbers.) Modeling Exercise 6.4.6 Experiment. Can you find other values for c and k that subject the rider to less acceleration while not bottoming out the shock in a 1.5 meter drop? modeled in Example 4.1) my
′′
(t)+cy
′
(t)+ky(t)=−mg,

Answer 1

The method of undetermined coefficients is used to find the particular solution to a second-order linear differential equation modeling the displacement of a shock absorber. The minimum value of the spring constant k and the damping coefficient c* that prevent bottoming out of the shock are determined. Further experimentation may lead to values of c and k that offer a less intense acceleration profile.

Step-by-step explanation:

Method of Undetermined Coefficients and Shock Displacement Analysis

To find the particular solution y_p(t) of the second-order linear differential equation given by my''(t) + cy'(t) + ky(t) = -mg, we use the method of undetermined coefficients. The particular solution will depend on the spring constant k. Moving forward, we apply the initial conditions y(0)=0 and y'(0)=-5.42 to find the displacement y(t) of the shock absorber. The displacement as a function of time involves the indeterminate k.

To determine the minimum value of k such that the shock does not compress more than -0.14 meters, a graphical or analytical method can be utilized. This also allows us to find the corresponding damping coefficient c*. After determining k* and c*, the displacement y(t) and the acceleration y''(t) are plotted over the time interval from 0≤t≤1 to assess the maximum acceleration experienced by the rider. The goal is to keep the acceleration within acceptable limits while not allowing the shock to bottom out, even when dropping from a height of 1.5 meters.

Further experimentation can yield different values of c and k that could potentially reduce the acceleration experienced by the rider without bottoming out the shock. This involves setting up and solving the system's dynamic equations considering the mass m, the gravitational constant g, and the parameters c and k.