Physics behind a tiny ping pong ball

What is slowing down the ball ??

Air Resistance (referred as drag in physics)acting on an object depends on multiple factors: 1. mass of object(m)    2. density of the object  (p)    3.  velocity that the object travels at(v)  4.  the cross sectional area that drag is acting on  (A) : which in this case is one half of the total surface area.

Shown below is the drag force on a travelling standardized ping pong ball with a mass of 2.7 grams and radius 20 mm

As we can see, the  formulate eventually reduces into a constant times the square of the velocity.

What other forces are acting on the ball?

As shown above, besides drag, force of gravity is also acting on the travelling ping pong ball.  we can based on the deceleration of the ball to calculate the x-component and the y-component.  

Notice that the drag always points the opposite of the direction that the object is travelling.  

By comparing the screen shots of the two ping pong ball motion analysis we can see  that since the balls were travelling at a different velocity, their deceleration due to drag is also different. For the same object, the drag depends on their velocity.

a = change in Velocity / change in time  

first ball's horizontal acceleration = - .96 m/sec/sec                      

second ball's horizontal acceleration= -1.78 m/sec /sec

As shown above, both the components can be found if know the drag, the velocity and mass of the object (ball in this case).  

Now Let's introduce a new topic:
the bouncing effect.

As shown below, an international standardized ping pong should bounce up 25 cm when dropped 30 cm above a standardized table tennis table.  

Based on the information, we can calculate the e (coefficient of restitution)

however, often times a ping pong ball does not land vertically, it lands obliquely. so we need to analyze the motion a little bit further !

In this case (Non-spinning) , the reflected angle is greater than the angle of impact; the vertical velocity becomes less than before, but the horizontal velocity stays the same.

However, practically the ping pong ball is rarely Non-spinning,  therefore a different approach is needed.

Because of the topspin , the bottom of the ball is traveling with either decreased velocity to the right, no net velocity, or velocity to the left. Three results are possible depending on the magnitude of vspin:

a. If vspin = uH the bottom of the ball has no net horizontal velocity and the horizontal force on impact is zero. As a result the horizontal velocity is unchanged.

b. If vspin < uH the horizontal velocity of the ball is reduces at the bottom. This subsequently reduces the horizontal force. The horizontal velocity will be reduced after impact but to a lesser degree that is experienced in the nonspinning case.

c. If vspin > uH the bottom of the ball will have a velocity which is opposite to the motion of the ball. Upon impact this will result in a force which increases the horizontal velocity!

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