# Follow up question regarding: Deceleration rate of objects of different mass but the same otherwise

My first question was answered here: Deceleration rate of objects of different mass but the same otherwise . But based on that, here is a follow-up question (I also edited my first post with the follow up question, but was not sure if that would be seen):

Follow up question (based on answer #1):

If it's true that the lighter ball would decelerate faster (and consequently take longer to travel a given distance), then what would be the difference in the initial velocity of the two balls (one 1 ounce, the other 2 ounces) if both are struck with the same implement with the same force (let's say a tennis racket travelling 100 mph).

I'm assuming the lighter ball would have a higher initial speed. If so, would the higher initial speed of the lighter ball offset the increased deceleration rate for the lighter ball. In practical terms: with the initial speed being different, which ball would arrive first in the above example of 100 feet, the lighter ball or the heavier ball?

If not too difficult, could you explain how this relationship (first the initial speed difference, and then the total travel time difference for 100 feet) would be calculated?

Here's a copy of my first question if you need it:

Using a tennis ball as an example object, if one ball weighs 1 ounce and the other is 2 ounces, and both are struck at 100 mph on the same trajectory, would there be any difference in the deceleration rate between the 2 different mass balls? (All other things about the two balls being equal). For instance, would the elapsed time for the ball to travel let's say 100 feet be different or the same?

## 1 Answer

Real world problems are hard, which is why the problems you see in texts are idealized. Look at this video of a tennis serve and you see nothing in the problem is rigid. The ball goes completely flat. Heavier balls (squash) distort the racquet greatly.

If you assume the balls are rigid and struck by a rigid, infinitely massive object, they will both leave at twice the object speed. If the object is just much more massive than the balls, the lighter one will be very slightly faster. You calculate this by writing the equations for energy and momentum before and after the collision. It is easy algebra. Let the incoming object have mass $M$ and velocity $V$ to start. It finishes with velocity $V'$ The tennis ball has mass $m$, initial velocity $0$, and final velocity $v'$ We have $$MV=MV'+mv'\\\frac 12MV^2=\frac 12M(V')^2+\frac 12m(v')^2\\MV^2=M(V')^2+m(v')^2\\(V+V')(V-V')=\frac mM(v')^2\\V-V'=\frac mMv'\\V+V'=v'\\v'=\frac {2V}{1+\frac mM}$$ so if $m \ll M, v' \approx 2V$ and it depends weakly on $\frac mM$

Nothing else is easy to calculate. The drag equation has the drag coefficient in it, which needs to me measured. The only reason your earlier question was easy was that the balls were traveling at the same speed, so presumably have the same drag coefficient. If you know how the drag coefficient depends on speed, you can just step through time, computing the acceleration, velocity, and position. Calculating the initial speed depends on the internal dynamics of the ball and racquet in impossible ways.

• Thank you Ross, that is extremely helpful. I really appreciate your time and explanation. Commented Jun 15, 2014 at 20:35
• Well, across the range of speeds a ball could achieve approximating the drag coefficient as constant would probably be good enough. Commented Oct 31, 2014 at 22:38