Initial Value Problem Applications - Motion under Gravity, Motion in a Medium

Motion in a Medium with Resistance

If the only force acting on a body is the resistance of the medium proportional to a power of the velocity v(t) the equation of motion becomes

x''(t)=a(t)=v'(t)=-v(t)^a

Assume the body starts at the origin, x=0 with unit initial velocity, v(0)=1. We have a number of natural questions to answer:

Will it move forever or stop at some value of time, t_stop? If travel time is infinite, will the distance travelled be infinite as well? How to find the distance and time travelled in case they are finite?

a =2 or a>2 ⇒Travels infinite distance in infinite time. Below a=2

a=1 or 1<a<2 ⇒ Travels finite distance in infinite time. Below a=1

0<a<1 ⇒ Travels finite distance in finite time. Below a=¼

Now do Mathematica practice

Motion in a medium with resistance

Motion Under Gravity

Launching a rocket under the variable gravitational force assumption leads to the following initial value problem with respect to v(r), the velocity as a function of distance to the Earth center

v(r)v'(r)=-GM/r², v(R)=a

where R is the radius of the Earth, G universal gravitational constant and a the initial velocity.

The exact solution is v(r)=(a²-2MG/R+2MG/r)^1/2