Chapter 11.1, Problem 54E

(a)

To determine

To calculate: The length of the path traveled by the projectile.

Answer to Problem 54E

The required length of the path traveled by the projectile is 641.236 feet.

Explanation of Solution

Given information:

The parametric equations: $x=\left(v_0\cos \theta \right)t$ , $y=\left(v_0\sin \theta \right)t-16t^2$

The angle: $\theta =30°$

The initial velocity: $v_0=150$

Calculation:

The given parametric equations are $x=\left(v_0\cos \theta \right)t$ and $y=\left(v_0\sin \theta \right)t-16t^2$

Differentiate the parametric equations with respect to t .

  $\frac{dx}{dt}=v_0\cos \theta$ , $\frac{dy}{dt}=v_0\sin \theta -32t$

At t = 0; $y=0$

Substitute 0 for y in the expression $y=\left(v_0\sin \theta \right)t-16t^2$ .

  $\begin{array}{c}0=\left(v_0\sin \theta \right)t-16t^2\\ t\left(v_0\sin \theta -16t\right)=0\end{array}$

Thus, $t=0$

And, $\begin{array}{c}{v}_0\sin \theta -16t=0\\ 16t=v_0\sin \theta \\ t=\frac{v_0\sin \theta }{t}\end{array}$

Consider that L be the length of the parametric curve.

So, $L={\displaystyle \underset{t_2}{\overset{t_2}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$

Write the length of the path traveled by the projectile as:

  $L={\displaystyle \underset{0}{\overset{\frac{v_0\sin \theta }{16}}{\int }}\sqrt{{\left(v_0\cos \theta \right)}^2+\left(v_0\sin \theta -32t^2\right)}dt}$

Also,

  $\begin{array}{c}\frac{dy}{dt}=0\\ v_0\sin \theta -32t=0\\ t=\frac{v_0\sin \theta }{32}\end{array}$

The maximum height at $t=\frac{v_0\sin \theta }{32}$ is:

  $y_{\text{max}}=\left(v_0\sin \theta \right)\left(\frac{v_0\sin \theta }{32}\right)-16{\left(\frac{v_0\sin \theta }{32}\right)}^2$

Now, substitute $30°$ for $\theta$ and 150 for $v_0$ in the expression $L={\displaystyle \underset{0}{\overset{\frac{v_0\sin \theta }{16}}{\int }}\sqrt{{\left(v_0\cos \theta \right)}^2+\left(v_0\sin \theta -32t^2\right)}dt}$ .

  $\begin{array}{c}L={\displaystyle \underset{0}{\overset{\frac{75\sin 30°}{8}}{\int }}\sqrt{{\left(150\cos 30°\right)}^2+\left(150\sin 30°-32t^2\right)}dt}\\ =641.236\end{array}$

Hence, the required length of the path traveled by the projectile is 641.236 feet.

(b)

To determine

To calculate: The maximum height of the projectile.

Answer to Problem 54E

The required maximum height of the projectile is 87.891 feet.

Explanation of Solution

Given information:

The parametric equations: $x=\left(v_0\cos \theta \right)t$ , $y=\left(v_0\sin \theta \right)t-16t^2$

The angle: $\theta =30°$

The initial velocity: $v_0=150$

Calculation:

The given parametric equations are $x=\left(v_0\cos \theta \right)t$ and $y=\left(v_0\sin \theta \right)t-16t^2$

Differentiate the parametric equations with respect to t .

  $\frac{dx}{dt}=v_0\cos \theta$ , $\frac{dy}{dt}=v_0\sin \theta -32t$

At t = 0; $y=0$

Substitute 0 for y in the expression $y=\left(v_0\sin \theta \right)t-16t^2$ .

  $\begin{array}{c}0=\left(v_0\sin \theta \right)t-16t^2\\ t\left(v_0\sin \theta -16t\right)=0\end{array}$

Thus, $t=0$

And, $\begin{array}{c}{v}_0\sin \theta -16t=0\\ 16t=v_0\sin \theta \\ t=\frac{v_0\sin \theta }{t}\end{array}$

Consider that L be the length of the parametric curve.

So, $L={\displaystyle \underset{t_2}{\overset{t_2}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$

Write the length of the path traveled by the projectile as:

  $L={\displaystyle \underset{0}{\overset{\frac{v_0\sin \theta }{16}}{\int }}\sqrt{{\left(v_0\cos \theta \right)}^2+\left(v_0\sin \theta -32t^2\right)}dt}$

Also,

  $\begin{array}{c}\frac{dy}{dt}=0\\ v_0\sin \theta -32t=0\\ t=\frac{v_0\sin \theta }{32}\end{array}$

The maximum height at $t=\frac{v_0\sin \theta }{32}$ is:

  $y_{\text{max}}=\left(v_0\sin \theta \right)\left(\frac{v_0\sin \theta }{32}\right)-16{\left(\frac{v_0\sin \theta }{32}\right)}^2$

Substitute $30°$ for $\theta$ and $150$ for $v_0$ in the expression $y_{\text{max}}=\left(v_0\sin \theta \right)\left(\frac{v_0\sin \theta }{32}\right)-16{\left(\frac{v_0\sin \theta }{32}\right)}^2$ .

  $\begin{array}{c}{y}_{\text{max}}=\left(150\sin 30°\right)\left(\frac{150\sin 30°}{32}\right)-16{\left(\frac{150\sin 30°}{32}\right)}^2\\ =87.891\end{array}$

Hence, the required maximum height of the projectile is 87.891 feet.