Range of mortar shells. The following are experimental data...

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}

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}

Key Concepts

Projectile Motion

Projectile motion is the study of the motion of an object launched into the air, where the only significant force acting on it (after launch) is gravity. This motion is characterized by a parabolic path, and it serves as a fundamental example in kinematics and dynamics in physics. The analysis typically assumes no air resistance, so the motion can be solved using basic formulas, allowing one to predict quantities such as the maximum height, time of flight, and horizontal range.

Kinematic Equations for Uniform Acceleration

Kinematic equations describe the relationships between displacement, velocity, acceleration, and time, especially when the acceleration is constant, as is the case under gravity. These equations provide the mathematical framework for predicting the motion of projectiles. In the context of a projectile fired at an angle, such as 45°, they help compute the time the object stays in the air and the distance it travels horizontally.

Time of Flight and Range for a Projectile

The time of flight and the range of a projectile are key quantities that can be determined using its initial velocity and launch angle, along with the acceleration due to gravity. When a projectile is fired at an optimal angle (often 45° in an idealized scenario), the range and flight time can be expressed in relatively simple formulas. This relationship highlights how the square of the initial velocity governs the range, and how the components of velocity determine the duration of flight.

Experimental Validation of Theoretical Models

Comparing experimental data with predictions from simple theoretical models is a common practice in physics. It validates the assumptions of the theory and may indicate where additional factors, such as air resistance or measurement errors, become significant. In the case of projectile motion, the experimental measurements of range and time of flight can be directly compared to the theoretical formulas derived from kinematics, revealing any regular patterns or discrepancies.

Trial Method for Solving Differential Equations

The trial method, also known as the method of undetermined coefficients or trial-and-error approach, is a technique for finding a particular solution to differential equations. By guessing a solution form based on the structure of the differential equation—often informed by physical insight—one can then adjust the coefficients to satisfy the equation. This method is particularly useful in solving linear differential equations where the forcing terms and the homogeneous solutions have standard forms.

Trigonometric Derivatives

The properties of the derivatives of trigonometric functions, such as sine and cosine, are fundamental in solving many differential equations that involve oscillatory behavior. For example, knowing that the derivative of sine is cosine and the second derivative of sine is negative sine allows one to recognize patterns in second order differential equations. These properties are not only mathematically elegant but also provide the basis for modeling periodic phenomena in various physical systems.

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