Explore BrainMass
Share

Projectile motion and differential equations

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Please follow all instructions and answer accordingly. See the attached file.

Consider a projectile of mass m

© BrainMass Inc. brainmass.com October 16, 2018, 9:14 pm ad1c9bdddf
https://brainmass.com/math/calculus-and-analysis/projectile-motion-and-differential-equations-176963

Attachments

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.

$2.19
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.

View Full Posting Details