Range of mortar shells. The following are experimental data on the range and muzzle velocity of mortar shells, all fired at $45^{\circ}$ to the horizontal. The time of flight is also included. Compare these ranges and times with the simple theory. Can you see any regularity? (Data from U.S. Department of Army, Firing Tables FT4.2-F-1, December 1954.) Use $g=32 \mathrm{ft} / \mathrm{s}^{2}$
\begin{tabular}{ccc}
\hline Muzle velocity, ft/s & Range, yd & Time, $s$ \\
tude of the electric field vector. Often the superseript zero
(0) on the $E$ is omitted if no ambiguity is introduced. The equation of motion is, from Eq. (3.20),
$$
\frac{d^{2} x}{d t^{2}}=\frac{q}{M} E_{x}=\frac{q}{M} E_{x}^{0} \sin \omega t
$$
In solving differential equations we shall often use the excellent method of trial and error, guided by physical insight. We look for a solution of the form $^{1}$
$$
x(t)=x_{1} \sin \omega t+v_{0} t+x_{0}
$$
On differentiating Eq. (3.46), we find
$$
\frac{d^{2} x}{d t^{2}}=-\omega^{2} x_{1} \sin \omega t
$$
The derivatives of the sine and cosine are $g$ ven by
$$
\begin{array}{ll}
\frac{d}{d \theta} \sin \theta=\cos \theta & \frac{d^{2}}{d \theta^{2}} \sin \theta=-\sin \theta \\
\frac{d}{d \theta} \cos \theta=-\sin \theta & \frac{d^{2}}{d \theta^{2}} \cos \theta=-\cos \theta
\end{array}
$$
\hline 334 & 1063 & $14.4$ \\
368 & 1268 & $15.7$ \\
400 & 1475 & $17.0$ \\
431 & 1683 & $18.2$
\end{tabular}