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, tstop? 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
This gives us the escape velocity (from Earth gravitational field, known as the second cosmic velocity) ve=(2MG/R)1/2≈11.186 km/s
Now do Mathematica practice
Air Resistance Makes Sky Diving Possible
Now do Mathematica practice
Sliding under Gravity
Now do Mathematica practice