Chapter 3.7, Problem 86E

Flight of a Rocket Suppose a rocket is fired upward from the surface of the earth with an initial velocity v (measured in meters per second). Then the maximum height h (in meters) reached by the rocket is given by the function

$h(v)=\frac{R{v}^2}{2gR-v^2}$

where R = 6.4 × 10⁶ m is the radius of the earth and g = 9.8 m/s² is the acceleration due to gravity. Use a graphing device to draw a graph of the function h. (Note that h and v must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

To determine

To sketch: The graph of maximum height cover by the rocket $h\left(v\right)=\frac{R{v}^2}{2gR-v^2}$ and determine the significance of vertical asymptote.

Explanation of Solution

The maximum height reached by the rocket is given below by the function,

$h\left(v\right)=\frac{R{v}^2}{2gR-v^2}$

Where h is the maximum height in meters, v is the initial velocity in meters per second, R is the radius of the earth and the value of it is $6.4\times {10}^6\text{ m}$ and g is the acceleration due to gravity which is $9.8{\text{ m/s}}^2$.

Substitute $6.4\times {10}^6\text{ m}$ for R and $9.8{\text{ m/s}}^2$ for g in given function, $h\left(v\right)$ the function becomes,

$\begin{array}{c}h\left(v\right)=\frac{\left(6.4\times {10}^6\right)\cdot v^2}{2\cdot 9.8\cdot \left(6.4\times {10}^6\right)-v^2}\\ =\frac{\left(6.4\times {10}^6\right)\cdot v^2}{125.44\times {10}^6-v^2}\end{array}$

Sketch the graph of above given rational function where h and v must be positive,

Figure (1)

The vertical asymptotes occur where the denominator is zero and where the function is undefined.

The zeros of denominator of rational function,

$\begin{array}{c}{v}^2=125440000\\ v=\pm \sqrt{125440000}\\ v=\pm 11200\end{array}$

Velocity must be positive value so the value of v is 11200.

The vertical asymptote is $v=11200$ where the function is undefined.