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Projectile motion and differential equations

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Consider a projectile of mass m

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Solution Summary

This shows how to complete a series of calculations, including velocity and projectile motion, governing equation of motion, second order differential equations, Bessel equations, Frobenius method, general solutions, transformation, and coupled systems.

Similar Posting

Deriving the equation of motion of a projectile shot vertically upward considering the effects of air resistance and solving the first order differential equation obtained.

Consider a projectile of mass m which is shot vertically upward from the surface of the earth with initial velocity V. Assume that the gravitational force acts downward at a constant acceleration g while the force of air resistance has a magnitude proportional to the square of the velocity with proportionality constant k>0 and acts to resist motion. Let x=x(t) denote the height of the projectile at time t and v(t) = dx/dt(t) , its velocity.

a) Explain why the governing equation of motion is given by:

mdv/dt = -kv^2 - mg v > 0 For t > 0 (1)
mdv/dt = kv^2 - mg v < 0

x(0) = 0 and v(0) = Vo

b) Solve this system as follows: Introduce V(x) = v[t(x)]. Then define V1(x) = V^2(x) for V(x) >0 and V2(x) = V^2(x) for V(x) < 0. Show that V1 and V2 satisfy

dV1/dx + 2k/m V1 = -2g , V1(0) = Vo^2
dV2/dx - 2k/m V1 = -2g , V2(Xm) = 0

Where Xm is defined implicitly by V1(Xm) = 0. Demonstrate that V2(0) < Vo^2.

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