Question

Projectile motion with air resistance differential equations

Answer 1

Obviously I'm asking because I don't know. More pertinently I can't imagine how to model this properly. Presumably we'll need independence on the x and y axes, connected with time. For free-fall I could model it pretty easily: m g − k v 2 = m ⋅ d v d t mg−kv2=m⋅dvdt 

Reference https://www.physicsforums.com/threads/projectile-motion-with-air-resistance.712807/

Answer 2

Suppose that a mass m is fired by a gun with an angle of θθ with the horizontal and suppose that the initial velocity of the mass is v0v0 feet per second. Neglect all forces except gravity and air resistance. Air resistance is equal to the kk ×× velocity of the object in (ft/sec).

1) Take the origin as the position of the gun and x-axis as horizontal and y-axis as vertical. Show that the differential equation of the resulting motion are

md2xdt2+kdxdt=0md2xdt2+kdxdt=0

md2ydt2+kdydt+mg=0md2ydt2+kdydt+mg=0

2) Find the solution fo the systems of differential equation in (1)

My work,

I have drawn a diagram and solve for (1). The motion of the object is upwards so air resistance is acting opposite to the motion of the object. Gravity is always downwards.

I am not sure how to do the (2) one.

(mD2+kD)x=0(mD2+kD)x=0

x=c1+c2e−ktmx=c1+c2e−ktm

(mD2+kD)y=−mg(mD2+kD)y=−mg


y=k1+k2e−ktm−mgtk