Solved: The Paris Guns The first mathematically correct theory of projectile motion was

Chapter 3, Problem 61

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The Paris Guns The first mathematically correct theory of projectile motion was originally formulated by Galileo Galilei (1564–1642), then clarified and extended by his younger collaborators Bonaventura 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 Torricelli did not have calculus at their disposal, so 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 \(\left\|\mathbf{v}_{0}\right\|=v_{0} \mathrm{\ m} / \mathrm{s}\) Let the projectile’s height above the ground be y meters and its horizontal distance from the launch site be x meters, and for convenience take the launch site to be the origin in the xy-plane. Then Galileo’s hypotheses can be represented by the following initial value problem:

\(\frac{d^{2} x}{d t^{2}}=0\)

\(\frac{d^{2} y}{d t^{2}}=-g\),

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 FIGURE 3.R.5 and Problem 23 in Exercises 3.12.

(a) Note that the system of equations in (2) is decoupled; that is, it consists of separate differential equations for x(t) and y(t). Moreover, each of these differential equations can be solved simply by antidifferentiating twice. Solve (2) 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 xy-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 gun be fired to maximize its range? The range is the horizontal distance traversed by a projectile before it hits the ground. Show that according to (2), 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 is launched with \(\theta=45^{\circ}\) for maximum range is \(v_{0}^{2} /(4 g)\).