In baseball, which can be hit further; a fast pitch or a slow pitch?

Factual Questions
1

This one came up the other day with our bar trivia group and we were unable to come up with a satisfactory answer. (It wasn’t one of the questions, it just came up since a game was on the TV while we were playing.)

Several people were arguing that the faster the ball was thrown, the further it could be hit. Their argument was that a faster ball would “bounce” off the bat with more force and so fly further (much as a faster ball will bounce further off a stationary object like a wall than a slower ball would.)

Others argued that the distance was more related to how hard the ball was hit. More energy would be lost from the swing in slowing down, stopping and reversing the direction of the faster ball and so it would be hit a shorter distance.

Our resident baseball experts disagreed with each other and Googling things like “fast ball distance” aren’t giving me a clear answer. I suspect it would depend on how elastic the collision between the ball and bat is but it’s been long enough since I tried to do that kind of calculation I couldn’t come up with anything. I also suspect that the answer is going to involve “It depends…”. So, can any Dopers better at the physics of baseball than we are help us out?

2

Of course a faster pitch would be hit farther. Instead of thinking of a moving ball hitting a moving bat, change your reference frame to one where the bat is stationary, and the ball comes in and bounces off it. A fast pitch will be travelling, relative to the bat, at maybe 170 mph, while a slower pitch will be travelling at 150 mph. Considering both of these are then bouncing off the bat, which one will come back with more speed? The faster pitch.

3

The fast ball would travel further, if hit correctly.

When the ball hits the bat, the kinetic energy of the ball’s motion is stored by compression of the ball (and to a lesser degree the bat). The ball then springs back into it’s original shape and travels off the bat in the opposite direction - assuming the ball hits the bat dead on.

It’s amazing how much balls deform when they are hit in sports. I’m trying to find some pics on google, but no luck so far.

4

Here’s a tennis ball gone squishy.

5

This is the rub (the bolded part). This subject came up a few months ago in GQ (I think).

In a perfect collision the faster pitch would travel out further. But a slower moving pitch is easier to hit with the sweet part of the bat. That is why batting practice pitches and home run derby contests HRs go so much further than game situation HRs (except for rare occasions when someone connects perfectly with a fastball or jumps all over a hanging curve)

6

You can take this even further: imagine a batter whose arms are amazingly strong, who will always hit uniformly. Now let’s line up our pitches:

  1. a 100m/s fastball pitched by a military-grade cannon
  2. a 50m/s “fastball” pitched by Nolan Ryan
  3. a regulation baseball sitting on a tee in front of the batter
  4. a ball thrown by a person standing behind the catcher, moving at 20m/s

Assume a near-elastic collision (because losses due to heat or air friction in the immediate vicinity of the collision are small) and use

Conservation of Momentum (mass x velocity):
m[sub]1[/sub]v[sub]1i[/sub] + m[sub]2[/sub]v[sub]2i[/sub] = m[sub]1[/sub]v[sub]1f[/sub] + m[sub]2[/sub]v[sub]2f[/sub]
where “m” is mass, “v” is velocity, subscripts “1” and “2” are “bat” and “ball”, and subscripts “i” and “f” are “initial” and “final”, and assuming no mass change in masses 1 or 2

and
Conservation of Kinetic Energy
[sup]1[/sup]/[sub]2[/sub] m[sub]1[/sub]v[sub]1i[/sub][sup]2[/sup] + [sup]1[/sup]/[sub]2[/sub] m[sub]2[/sub]v[sub]2i[/sub][sup]2[/sup] = [sup]1[/sup]/[sub]2[/sub] m[sub]1[/sub]v[sub]1f[/sub][sup]2[/sup] + [sup]1[/sup]/[sub]2[/sub] m[sub]2[/sub]v[sub]2f[/sub][sup]2[/sup]

I assume a bat of mass 5 kilograms (probably too light) and a ball of .5kg (probably too heavy), with a swing speed of 100m/s (probably too fast) coming into the ball. The ball speeds are in cases 1-4 above, but realize that for the first two, we prepend a negative sign to indicate that they’re moving in a direction opposite that of the bat.

Conservation of Momentum:
500 kg*m/s + {-50, -25, 0, or 10} = 5v[sub]1f[/sub] + .5v[sub]2f[/sub]
and so for Case I: v[sub]1f[/sub] = (450 - .5v[sub]2f[/sub]) / 5
Case II: v[sub]1f[/sub] = (475 - .5v[sub]2f[/sub])/5
Case III: v[sub]1f[/sub] = (500 - .5v[sub]2f[/sub])/5
Case IV: v[sub]1f[/sub] = (510 - .5v[sub]2f[/sub])/5

For each case, we’ve now expressed v[sub]1f[/sub] as a function of v[sub]2f[/sub], which we can substitute into the equations for

Conservation of Mass:
25,000 + .25 * {10,000 ; 2,500 ; 0 ; or 400} = 2.5 * (v[sub]1f[/sub])[sup]2[/sup] + .25 * (v[sub]2f[/sub])[sup]2[/sup]

Substituting from above, we have for Case I:
25,000 + 2,500 = 2.5 * {(475 - .5v[sub]2f[/sub])/5}[sup]2[/sup] + .25 * (v[sub]2f[/sub])[sup]2[/sup]

Case II: 25,000 + 625 = 2.5 * {(450 - .5v[sub]2f[/sub])/5}[sup]2[/sup] + .25 * (v[sub]2f[/sub])[sup]2[/sup]
Case III: 25,000 = 2.5 * {(500 - .5v[sub]2f[/sub])/5}[sup]2[/sup] + .25 * (v[sub]2f[/sub])[sup]2[/sup]
Case IV: 25,000 + 100 = 2.5 * {(510 - .5v[sub]2f[/sub])/5}[sup]2[/sup] + .25 * (v[sub]2f[/sub])[sup]2[/sup]

So now we’ve reduced to solvable quadratics. I dump these into Excel (my trusty graphing calculator being otherwise occupied) and get… a bunch of gibberish. I’m going to leave this posted and try a different solution.

When come back, bring roots.

7

Wow, impressive pile of equations, Jurph, though I like CurtC’s simple thought experiment better. Actually, you don’t really need all the heavy artillary; another way to express CurtcC’s model is to say that if the bat’s motion and contact with the ball is identical in both cases, that you have conservation of momentum, and a ball that starts out faster is going to end up going faster. The relationship may not be linear due to energy loss from elasticity effects, but it will go faster.
Fear has a good explanation of why our intuition may not match the physics.

8

Surprisingly, I read in a recent Natural History magazine that a slower moving curveball can be hit further than a faster moving (naturally) fastball. This has to do with the lift created by the rapid spin of a curveball, or something like that. I’ll try to find the article and get into more specifics later.

9

I remember that article, but couldn’t remember where I had seen it. I remember, as you do, that the difference in distance due to the spin of the ball was greater than any potential difference due to the speed of the ball.

10

Just a few small facts to throw into the equation (and I look forward to seeing the results).

The weight in ounces of a standard bat varies between 26 and 36 roughly. So the 5Kg is way to high.

I believe (but don’t have the time to check) that a standard baseball weighs between 5 and 5.25 ounces.

Also, just in case it’s relevant, the size of a bat is usually between 28 and 34 inches.

11

Check out this article.

  • Hmm, what about a slider? :wink:

Food for thought

12

Thanks for the feedback all. I think Fear has the reason we were confusing ourselves. I had made the analogy of the ball bouncing off a wall to suggest the fast ball should go further but our more baseball savvy members had said that a slow, “hanging” ball would be hit further. We were trying to analyze the physics of the situation without considering the fact that the slower ball was simply easier to hit. (Proving once again that it is possible to overanalyze a problem :smack: )

I think I was overestimating the amount of energy lost in the ball-bat collision too.

(And thanks for the equation work Jurph. Man, it has been a long time since I tried working something like that out. Don’t use momentum and kinetic energy much in statistical forecasting.)

13

Empirical data would probably be more useful … This is mostly anecdotal, but you’re welcome to it. I spent a deplorable portion of my childhood watching the great Lee Smith closing games for the Cubs. He was a pure fireballer-threw almost nothing but fastballs, pretty consistently in the 95-100 MPH range. And one of the things I noticed was that, while batters rarely connected solidly with these pitches, when somebody did manage to get around on it and hit it with the ‘sweet spot’, the correct procedure for the religious Cubs fan was to pray for a foul ball. 'Cause God Himself had no chance of keeping that thing inside the park.