Problem 59 - Advanced Engineering Mathematics

Question

The Paris Guns The first mathematically correct theory of projectile motion was originally formulated by Galileo Galilei (1564-1642), then clarified and extended by bis younger collaborators Bonaventara Cavalieri (1598-1647) and Evangelista Torricelli (1608-1647). Galileo's theory was based on two simple hypotheses suggested by experimental observations: that a projectile moves with constant horizontal velocity and with constant downward vertical acceleration. Galileo, Cavalieri, and Toricelli did not have calculus at their disposal, 80 their arguments were largely geometric, but we can reproduce their results using a system of differential equations. Suppose that a projectile is launched from ground level at an angle \(\theta\) with respect to the horizontal and with an initial velocity of magnitude \(\left\|v_{0}\right\|=v_{0} \mathrm{~m} / \mathrm{s}\). Let the projectile's height above the ground be \(y\) meters and its harizontal distance from the launch site be \(x\) meters, and for convenience take the launch site to be the origin in the \(x y\)-plane. Then Galileo's hypotheses can be represented by the following initial-value problem: $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}=0 \\ &\frac{d^{2} y}{d t^{2}}=-g \end{aligned} $$ where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}, x(0)=0, y(0)=0, x^{\prime}(0)=v_{0} \cos \theta\) (the \(x\)-component of the initial velocity), and \(y^{\prime}(0)=v_{0} \sin \theta\) (the \(y\)-component of the initial velocity). See FICURE \(3 . R .4\) and Problem 23 in Exercises 3.12. FICURE 3.R.4 Ballistic projectile (a) Note that the system of equations in (1) is decoupled, that is, itconsists of separate differential equations for \(x(t)\) and \(y(t)\). Moreover, each of these differential equations can be solved simply by antidifferentiating twice. Solve (1) to obtain explicit formulas for \(x(t)\) and \(y(t)\) in terms of \(v_{0}\) and \(\theta\). Then algebraically eliminate \(t\) to show that the trajectory of the projectile in the \(x y\)-plane is a parabola. (b) A central question throughout the history of ballistics has been this: Given a gun that fires a projectile with a certain initial speed \(v_{0}\), at what angle with respect to the horizontal should the gunbe fired to maximize its range? The range is the horizontal distance traversed by a projectile before it hits the ground. Show that according to (1), the range of the projectile is \(\left(v_{0}^{2} / g\right) \sin 2 \theta\), so that a maximum range \(v_{0}^{2} / g\) is achieved for the launch angle \(\theta=\pi / 4=45^{\circ}\). (c) Show that the maximum height attained by the projectile if launched with \(\theta=45^{\circ}\) for maximum range is \(v_{0}^{2} /(4 \mathrm{~g})\).

Step-by-Step Solution

Answer

The projectile follows a parabolic path; max range occurs at 45°, max height is \(v_0^2/(4g)\) at 45°.

Step 1: Solve for Horizontal Motion x(t)

From the equation \( \frac{d^2 x}{dt^2} = 0 \), we integrate once to get \( \frac{dx}{dt} = C_1 \). Using the initial condition \( x'(0) = v_0 \cos \theta \), we find \( C_1 = v_0 \cos \theta \). Integrating again, \( x(t) = v_0 \cos \theta \cdot t + C_2 \) and using \( x(0) = 0 \), we have \( C_2 = 0 \). Thus, \( x(t) = v_0 \cos \theta \cdot t \).

Step 2: Solve for Vertical Motion y(t)

From the equation \( \frac{d^2 y}{dt^2} = -g \), integrate once to get \( \frac{dy}{dt} = -gt + C_3 \). Using \( y'(0) = v_0 \sin \theta \), we find \( C_3 = v_0 \sin \theta \). Integrating again, \( y(t) = -\frac{1}{2}gt^2 + v_0 \sin \theta \cdot t + C_4 \). Using \( y(0) = 0 \), we find \( C_4 = 0 \), so \( y(t) = -\frac{1}{2}gt^2 + v_0 \sin \theta \cdot t \).

Step 3: Eliminate Time t to Find Trajectory

Solve for \( t \) in terms of \( x \) from \( x(t) = v_0 \cos \theta \cdot t \), which gives \( t = \frac{x}{v_0 \cos \theta} \). Substitute in \( y(t) \) to get the trajectory equation:\( y = -\frac{1}{2}g \left( \frac{x}{v_0 \cos \theta} \right)^2 + v_0 \sin \theta \cdot \frac{x}{v_0 \cos \theta} \).Simplifying, this yields:\( y = x \tan \theta - \frac{gx^2}{2v_0^2 \cos^2 \theta} \), confirming a parabolic path.

Step 4: Find Maximum Range

The range \( R \) is the value of \( x \) when \( y = 0 \). From \( y = x \tan \theta - \frac{gx^2}{2v_0^2 \cos^2 \theta} \), set \( y = 0 \) to solve for \( x \):\[ 0 = x \tan \theta - \frac{gx^2}{2v_0^2 \cos^2 \theta} \].Solving gives \( x = \frac{v_0^2 \sin 2\theta}{g} \), with maximum range when \( \sin 2\theta = 1 \), i.e., \( \theta = 45^\circ \).

Step 5: Find Maximum Height for θ = 45°

For maximum height when \( \theta = 45^\circ \), the vertical motion equation is considered at peak (where \( \frac{dy}{dt} = 0 \)):\[ 0 = -g \cdot t_{peak} + v_0 \sin 45^\circ \].Solve for \( t_{peak} = \frac{v_0 \sin 45^\circ}{g} \). Substitute \( t_{peak} \) in \( y(t) \) to find:\[ y_{max} = v_0 \sin 45^\circ \cdot \frac{v_0 \sin 45^\circ}{g} - \frac{1}{2}g \left(\frac{v_0 \sin 45^\circ}{g}\right)^2 \].This simplifies to \( \frac{v_0^2}{4g} \).

Key Concepts

Differential Equations in Projectile Motion

Differential equations are essential tools in physics, used to describe the behavior of dynamic systems, such as the motion of a projectile. In this context, differential equations help us understand how a projectile's position changes over time based on its initial conditions and external forces, like gravity.

For projectile motion, we typically deal with two main equations. One for horizontal movement and another for vertical movement. The horizontal differential equation is given by \(\frac{d^2 x}{dt^2} = 0\), indicating uniform motion without acceleration along the x-axis. This simplifies through integration to describe how horizontal velocity remains constant.

Meanwhile, the vertical motion equation is \(\frac{d^2 y}{dt^2} = -g\), reflecting the constant acceleration due to gravity. Integrating this twice, it encapsulates the effect of gravity on the projectile, showing how its vertical velocity decreases over time.

This breakdown elucidates how differential equations serve as a bridge between forces acting on the object and its resulting motion.

Galileo's Projectile Theory

Galileo's groundbreaking work laid the foundation for modern physics by introducing the theory of projectile motion. His insights were based on the idea that a projectile has constant horizontal velocity and a constant downward vertical acceleration.

To plot the motion, Galileo's theory uses geometric arguments rather than calculus, which wasn't available during his time. He visualized motion by observing the separate horizontal and vertical components, leading to the understanding of the parabolic trajectory of projectiles.

This theory proved instrumental because, despite its simplicity, it accurately described real-world motion without requiring complex calculations. It established the principle that forces impact objects along different axes independently, a crucial aspect in physics that allows for precise predictions of an object's path.

Maximum Range Calculation

The maximum range of a projectile is achieved when it travels as far as possible horizontally. The solution involves finding the optimal launch angle, which maximizes this horizontal distance before hitting the ground.

Mathematically, the range \(R\) can be expressed as \(R = \frac{v_0^2 \sin 2\theta}{g}\), where \(v_0\) is the initial velocity, \(\theta\) is the launch angle, and \(g\) is the acceleration due to gravity.

To maximize \(R\), \(\sin 2\theta\) should be at its peak, which occurs when \(\theta = 45^\circ\). Therefore, shooting a projectile at this angle ensures the maximum horizontal distance.

This elegant result demonstrates how the interplay of initial velocity and angle affects projectile paths, an insight crucial for fields ranging from engineering to sports.

Geometric Arguments in Physics

In physics, geometric arguments often simplify complex ideas without needing intensive calculations. They focus on using shapes and visual reasoning to understand physical laws.

Galileo's work beautifully demonstrates this approach by using simple geometric concepts to describe the path of a projectile. By considering the parabolic arc traced by the projectile, it became apparent how horizontal and vertical motions combine.

This method involves recognizing the symmetry and simplicity in physical systems. It is particularly useful in teaching, as it builds intuitive understanding and lays the groundwork for more advanced studies. These arguments reveal the natural order in physical relationships that might seem convoluted otherwise.

By focusing on geometry, learners can grasp the underlying principles of motion, facilitating a deeper understanding of how different forces shape trajectories.