Perform three iterations of the Newton-Raphson method to determine the root of Eq. (E7.1.1...

Perform three iterations of the Newton-Raphson method to determine the root of Eq. (E7.1.1). Use the parameter values from Example 7.1 along with an initial guess of t = 3 s.

Equation E7.1.1:

Example 7.1

Determining the Optimum Analytically by Root Location

Problem Statement. Determine the time and magnitude of the peak elevation based on Eq. (7.1). Use the following parameter values for your calculation: g = 9.81 m/s², z₀ = 100 m, υ₀ = 55 m/s, m = 80 kg, and c = 15 kg/s.

Solution. Equation (7.1) can be differentiated to give

 (E7.1.1)

Note that because υ = dz/dt, this is actually the equation for the velocity. The maximum elevation occurs at the value of t that drives this equation to zero. Thus, the problem amounts to determining the root. For this case, this can be accomplished by setting the derivative to zero and solving Eq. (E7.1.1) analytically for

Substituting the parameters gives

This value along with the parameters can then be substituted into Eq. (7.1) to compute the maximum elevation as

We can verify that the result is a maximum by differentiating Eq. (E7.1.1) to obtain the second derivative

The fact that the second derivative is negative tells us that we have a maximum. Further, the result makes physical sense since the acceleration should be solely equal to the force of gravity at the maximum when the vertical velocity (and hence drag) is zero.

Although an analytical solution was possible for this case, we could have obtained the same result using the root-location methods described in Chaps. 5 and 6. This will be left as a homework exercise.

Equation (7.1):