Trajectory of projectile with considerable drag

[HP007]

Facing some horrible mathematical situation while solving to find equation of trajectory of projectile when drag is proportional to v^2.
my equations where i end up with are as follow:
equation 1:
mdv/dt=(-kv^2)+(-mgsinγ);
equation 2:
(-mv)dγ/dt=mgcosγ;
equation 3:
dx/dt=vcosγ;
equation 4:
dy/dt=vsiny;
where:
v is velocity of particle at instance when it makes an angle γ with horizontal plane.
Initial condition is known and assume it to be u at an angle α.
please assist me in solving this.

 

[Simon Bridge]

(1) $m\dot v = -kv^2-mg\sin\gamma$
(2) $-mv\dot \gamma = mg\cos\gamma$
(3) $\dot x = v\cos\gamma$
(4) $\dot y = v\sin\gamma$

Whence $v(0)=u$ and $\gamma(0)=\alpha$

Do you want the trajectory: (x(t),y(t))?

Have you ever tried to solve systems of differential equations before?
(i.e. what is the level of education help should be aimed at?)

Have you seen:
http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/mechan/air0.pdf

You seem to be trying to use cartesian and some sort of polar coordinates at the same time - it is best practice to pick just one coordinate system and stick to it.

 

[Nugatory]

This problem has come up before. Try this thread: https://www.physicsforums.com/showthread.php?t=712807

 

[HP007]

This problem has come up before. Try this thread: https://www.physicsforums.com/showthread.php?t=712807

But I have already arrived at those set of coupled equations I want to know the way to solve it.
Assist me in solving mathematical part of the problem.

 

[PJC01]

I understand the frustration and challenges that come with solving complex mathematical equations, especially when they involve variables that are not easily measurable. In the case of finding the trajectory of a projectile with considerable drag, the equations you have provided are a good starting point.

In order to solve these equations, you will need to use numerical methods or approximation techniques, as finding an analytical solution may not be possible. One approach is to use numerical integration methods, such as the Euler method, to approximate the trajectory of the projectile. This involves breaking down the equations into smaller time intervals and using iterative calculations to estimate the position and velocity of the projectile at each time step.

Another approach is to use software or programming languages, such as MATLAB or Python, to create a simulation of the projectile's motion. This will allow you to input the initial conditions and the equations, and then visualize the trajectory of the projectile. You can also adjust the parameters, such as the drag coefficient, to see how it affects the trajectory.

It is also important to consider the assumptions made in the equations, such as a constant drag coefficient and a uniform gravitational field. These assumptions may not hold true in real-world situations, so it is important to validate the results of your calculations with experimental data.

Overall, finding the equation of trajectory for a projectile with considerable drag can be a challenging task, but with the right tools and techniques, it is possible to get an accurate estimation of its motion. Keep in mind that the final solution may not be a simple equation, but rather a series of calculations or a simulation that can provide valuable insights into the behavior of the projectile.