Forest UCM PnCP ProjMotion
Friction depends linearly on velocity
Projectile motion describes the path a mass moving in two dimensions. An example of which is the motion of a projectile shot out of a cannon with an initial velocitywith an angle of inclination .
When the motion in each dimension is independent, the kinematics are separable giving you two equations of motion that depend on the same time.
Using our solutions for the horizontal and vertical motion when friction depends linearly on velocity (Forest_UCM_PnCP_LinAirRes) we can write :
wherehas replaced so the components are more explicitly identifiable.
in the y-direction however, the directions are changed to represent an object moving upwards instead of falling
Newton's second law for falling
for a rising projectile
This changes the signs in front of theterms such that
wherehas replced so the components are more explicitly identifiable.
We now have a system governed by the following system of two equations
To determine how far the projectile will travel in the x-direction (Range) you can solve the above equation forin the case that .
since time is the same in both equations you can solve for time in terms of x and substitute for time inthe y-direction equations.
solving forusing the x-direction equation
now we need to substitute for time
substituting for time
The Range is defined as the value for when
The above equation does not have an exact analytical solution.
You can try to solve it graphically or by taylor expanding small quantities when they appear as arguments to functions like thefunction
Solution by Taylor expansion
ln(1-x) Taylor expansion
Taylor expanding about x = 0 for
Taylor expanding about x=0
Taylor expand ln term in Range equation
- is a solution height is zero when