Chapter 8.FOM, Problem 1P

Trajectories Are Parabolas From the graphs in Figure 3 the paths of projectiles appear to be parabolas that open downward. Eliminate the parameter r from the general parametric equations to verify that these are indeed parabolas.

Figure $\left(3\right)$ Paths of projectiles

To determine

To verify:

The parametric equations are equation of parabola.

Answer to Problem 1P

Solution:

The given set of parametric equation is equation of parabola.

Explanation of Solution

Approach:

Eliminate the parameter $t$ from the parametric equation and find the equation in rectangular coordinates.

Calculation:

Consider the following parametric equations for a parabola opening downward.

$x=\left(v_0\cos \theta \right)t$ …$\left(1\right)$

$y=\left(v_0\sin \theta \right)t-\frac{1}{2}g{t}^2$ …$\left(2\right)$

Rewrite equation $\left(1\right)$ in terms of $t$.

$t=\frac{x}{v_0\cos \theta }$

Substitute $\left(\frac{x}{v_0\cos \theta }\right)$ for $t$ in equation $\left(2\right)$.

$\begin{array}{l}y=\left(v_0\sin \theta \right)\left(\frac{x}{v_0\cos \theta }\right)-\frac{1}{2}g{\left(\frac{x}{v_0\cos \theta }\right)}^2\\ y=x\tan \theta -\frac{1}{2}g\left(\frac{x^2}{v_0^2{\cos }^2\theta }\right)\end{array}$

Since the above equation is of the form $y=a{x}^2-bx+c$, therefore, the given set of parametric equation is the equation of a parabola.

Conclusion:

Hence, the given set of parametric equation is equation of parabola.