# B Physics of table tennis

1. Aug 6, 2017

### vishnu 73

i was watching a video of physics of table tennis

and got interested to work it out for myself

forces to be considered

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

2. Aug 6, 2017

### Staff: Mentor

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.

3. Aug 6, 2017

### vishnu 73

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?

4. Aug 6, 2017

### Staff: Mentor

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

5. Aug 7, 2017

### vishnu 73

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

6. Aug 7, 2017

### Staff: Mentor

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.

7. Aug 7, 2017

### vishnu 73

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

8. Aug 8, 2017

### Staff: Mentor

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.