A ship fires an explosive projectile with a weight W, from an initial angle θi from the horizontal and the initial velocity Vi, to hit a target. You are to write a program that will provide the solution of the motion of the projectile, that gives a table of values of x, y, Vx, Vy, and V ={Vx² + Vy²}

Question

A ship fires an explosive projectile with a weight W, from an initial angle [latex]θ_{i}[/latex] from the horizontal and the initial velocity [latex]V_{i}[/latex], to hit a target. You are to write a program that will provide the solution of the motion of the projectile, that gives a table of values of x, y, [latex]V_{x}, V_{y},[/latex] and V ={V[latex]^{2}_{x}[/latex] + V[latex]^{2}_{y}[/latex]}[latex]^{{1}/{2}}[/latex] at a number of even-spaced time increments, for any selected values of θ[latex]_{i}[/latex] and V[latex]_{i}[/latex]. Assume that, the air drag on the projectile will vary as the square of the total velocity, i.e., [latex]F_{drag} = KV_{2}[/latex]. The ODEs that describe the projectile motion are given below. Give the input your program would need to solve the motion if the weight of the projectile W = 400 lbs, V[latex]_{i}[/latex] = 800 ft/sec, θ[latex]_{i}[/latex] = 45°, and K = 0.0022.

Accepted Answer

To solve this problem, Newton’s Second Law of motion must be used in both the x and y directions giving the two simultaneous second-order ODEs given below.

Newton’s Second Law

∑[latex]F_{x} = ma_{x}[/latex] ∑[latex]F_{y} = ma_{y} [/latex]

−KV² cosθ= [latex]\frac{W}{g}\frac{dV_{x}}{dt}[/latex] - W -KV² sinθ= [latex]\frac{W}{g}\frac{dV_{y}}{dt}[/latex]

[latex]\frac{d^{2}x}{dt^{2}}= \frac{dV_{x}}{dt}= - \frac{gKVV_{x}}{W} \frac{d^{2}y}{dt^{2}}= \frac{dV_{y}}{dt}=-g - \frac{gKVV_{y}}{W}[/latex]

These second-order equations need to be written as two first-order equations, or the first becomes [latex]V_{x}[/latex] = dx/dt, with d[latex]V_{x}[/latex]/dt as given above, and the second produces the two equations [latex]V_{y}[/latex] = dy/dt and d[latex]V_{y}[/latex] /dt, as given above. Thus, in solving these four simultaneous ODEs, a solution is sought for x, [latex]V_{x}[/latex] , y, and [latex]V_{y}[/latex]. These unknowns will be contained in an array X of four elements, so that X(1) = x, X(2) = [latex]V_{x}[/latex], X(3) = y, and X(4) = [latex]V_{y}[/latex]. The derivatives that must be returned by the subroutine it calls on will also consist of four values, e.g., DXT(1) = dx/dt = [latex]V_{x}[/latex], DXT(2) = d[latex]V_{x}[/latex]/dt, DXT(3) = dy/dt = [latex]V_{y}[/latex], and DXT(4) = d[latex]V_{y}[/latex]/dt. The two listings below solve this problem by calling on RUKUST and RKFEHLSG, respectively. The input to these programs should consist of: 400 .0022 32.2 800 45 .1

Question C.2: A ship fires an explosive projectile with a weight W, from a...

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