Mastering the Physics of Table Tennis: Understanding Forces and Flight Dynamics

[timetraveller123]

i was watching a video of physics of table tennis

and got interested to work it out for myself

forces to be considered

quadratic drag force
magnus force
gravity

m : mass of the ball
v : velocity of the ball
w : angular velocity of the ball(assuming for simplicity not changing during the fllight)
r : radius of the ball
$\rho$ : density of air
g : gravitational acceleration
A cross sectional area of ball force on the ball

$-mg \hat j\\ \frac{c_d A \rho v^2}{2}\text {opposite to direction of velocity }\\ \frac{1}{2}{c_m \rho A v^2 \hat w \times \hat v}\\$
through out the flight the direction of magnus force and drag force are constantly changing and that is giving me some trouble setting up the f = ma differential equation
i plan to break the forces down into their vertical and horizontal forces and set up two differential equation to get y(t) and x(t) can it be done please help thanks

 

[mfb]

The differential equations will be coupled, and in general there is no proper analytic solution even without Magnus force.
You can always calculate numerical solutions, of course.

 

[timetraveller123]

i realized it would be coupled differential but how do i get to that please guide me to get the equation

and why do say there would be no analytic solution?

 

[mfb]

i realized it would be coupled differential but how do i get to that please guide me to get the equation

Put the three equations you have already into a single formula $m \vec a =$ and you are done. In general you have three coordinates to consider, as the ball can have side-spin ($\omega$ along the vertical axis).

and why do say there would be no analytic solution?

That's life. Analytic solutions for real problems are rare.

 

[timetraveller123]

then why it that in the video she just plugged the forces into the Lagrangian and got somewhat a analytical solution i didn't understand what she was doing at the last part
firstly i thought the Lagrangian cannot be used when there is non-conservative force here the drag and magnus is non conservative
secondly my understanding of the langrangian is that
$L = T - U$
in the video she just plugged in the forces instead of the energies into the lagrangian

 

[mfb]

An analytic solution for the position as function of time? Where? She gets the equations of motion, and then goes to numerical simulations.
Where do you see a Lagrangian?

She plugged in forces in ma=F.

 

[timetraveller123]

wait what so the equation of motion is only at one instant in time

at 5:27 she starts using the langrangian and i think what follows is the euler lagrange
isn't that the lagrangian

 

[mfb]

The equation of motion gives you the acceleration at every point in time if you know the velocity at this time.

@5:27: Sort of, but as far as I can see it is only used to motivate the equation of motion, not for calculations.