Why does a ping pong ball change direction when I spin it on a table? When I spin a ping pong ball on the table, it rolls forward in the opposite direction of the spin, and then eventually changes direction and rolls backward. 
Here's a video demonstrating the effect.
Why does that happen? 
 A: One way to explain it that makes sense to me is that the backwards spin on the ball is a force pushing the ball toward you. The one time applied force of your finger that creates the backward spin also pushes the ball forward, but the backspin stays almost constant. At the moment you spin the ball, the forward force is greater then that of the backspin, so the ball moves away. After time zero, the backspin is the only force acting horizontally (you aren't still touching the ball, so there is no applied force) and so the ball slows, then returns because of a negative acceleration.
If you were to graph this motion as Position (away from you) vs Time, you would get a graph that looks similar to the motion of a ball thrown upwards and pulled back by gravity. 
A: The ball moves forward when the force you applied is strong enough to overcome the friction of the table, but spins back towards you once that force degrades and the surface of the ball catches onto the surface of the table.
A: The problem is exactly analogous to a very famous problem which is presnt in any good Rotational book,
Whats happening here is ball has a velocity of its centre of mass,Vcm, and an angular velocity.
Now when we put the ball on a friction ON surface, the force of friction provides a torque and an acceleration to the centre of mass, this happens till the time the Vcm adn omega are related as Vcm = omega *R 
(R is the radius of the sphere)
Now when this happens the sphere begins to start pure rolling and friction ceases to act,
So initially say Vcm is greater than angular velocity,
then friction acts in direction opposite to Vcm and decelerates the body decreasing Vcm but at the same time it provides a torque to the body which increases its angular velocity,
this goes on till the 2 of them are equal,(Vcm and R*angular speed)
so the ball initially slips forward but after some time starts rolling backwards.
End of the story,
LOL
A: Because the spin of the ball is not, in such a case, aligned perfectly with the table. If the ball were spinning so that the axis of rotation were exactly perpendicular to the table, then the ball would not move anywhere.
Imagine the tire of a car for example. The axis of rotation of the tire is parallel with the ground, so the outer surface of the tire pushes against the ground.
Same thing with the ball: if the axis of rotation is not perfectly perpendicular to the surface, then the ball will push on the table like the wheel does.
However, the axis of rotation of the ball is nowhere near parallel with the table, so it only pushes a tiny bit.
Spin the ball with an exaggeratedly parallel axis and it will tend to roll more than spin.
EDIT: In the case of the video, the axis of rotation is parallel with the table. The ball will tend to go in the opposite direction of which the bottom surface of the ball pushes, just like the tires on a car.
