Can a baseball be hit farther with a heavy bat or a light one?

Cecil's Columns/Staff Reports
1

I believe your response was incorrect:
“The law of physics governing baseball bats, and a lot of other things, is:
F = ma
Where F= force resulting, m=mass and a=acceleration.”

This law does govern many things, however I think it’s not applicable in the baseball case. The way I think of a bat hitting a ball, there would be little or no acceleration. The bat is basically a massive object traveling with a velocity and making contact with the ball. Most of the acceleration is at the beginning of the swing to bring the bat up to speed, plus a slight inertial force because the bat is moving in a circle (but that doesn’t matter much in this case). I claim that the law governing the interaction is
E= 1/2 m v^2
where
E = kinetic energy of the bat
m = mass
v = velocity

If one ignores the energy lost due to heat and sound, etc, and assumes that all of the energy is transferred to kinetic energy of the ball at the moment of impact, then (since the ball has fixed mass) the energy would be in the form of velocity.

So, all other things being equal, if you double the speed of the bat, the ball will go 2 times as fast, but if you double the mass of the bat, the ball will go 1.414 times as as fast (square root of two).

However, after collision, some KE remains in the bat, so the numbers (2 and 1.4) would be slightly less.

But you had the right answer anyway, “So to answer your question, a baseball can be hit farthest with a heavy bat, assuming the game were played strictly by laboratory robots”, I just thought your reasoning was incomplete.

2

Both energy (.5 mv^2) and momentum (mv) come into all collision analysis.

But another factor that comes into this is the maximum speed at which the arms in question can drive the bat. For a very light bat, the mass of the arms and the response of the muscles becomes a limiting factor; for a very heavy bat, the mass of the bat becomes the main factor.

In other words, the correct answer is: “A medium-weight bat is best, where the definition of “medium” depends on the batter.”

3

nope - momentum is of critical issue here. If the speed of the two bats is identical, (which is assumed by the way the question is phrased) the bat with more momentum is going to hit the ball farther. You go ahead and try swinging a big bat at a ball and then swing a breadknife at a ball at the same speed. The heavy item imparts more force to the ball, and it goes farther. Trouble with physics majors is that while they’re home studying formulae, the rest of the kids are outside, playing ball. Einstein, by the way, is purported to have said that he learned all the physics he needed to know by the time he was three years old. That was his point, too.

4

The Mailbag Article is Can a baseball be hit farther with a heavy bat or a light one?

The book The Physics of Baseball by Robert K. Adair, has an interesting discussion of bat weights. Babe Ruth used bats from 56 oz. (early in his career) to 36 oz. (late in his career). He was using mostly a 47-oz. bat when he hit 60 homers in 1927. Roger Maris was using a 33-oz. bat when he hit 61 in '61. Hank Aaron used a 31 or 32 oz. bat.

In real life there are advantages to using a lighter bat that offset the slight distance advantage of a heavier bat. With the lighter bat, you can commit to the swing later and still contact the ball, and you have greater control over the angle at which bat and ball meet.

5

Technically, F=ma does apply. In fact, there is a very large acceleration. However, the time of the acceleration (to the ball) is very small (instantaneous). Thus it is not very useful for this case. Though it could be used for determining how fast the bat is moving, and thus calculating the bat’s kinetic energy and momentum.

CC, the question is specified in terms of fixed torque, not fixed velocity. Accordingly, the lighter bat will be moving faster.

The real question, the practical question that people are wanting to know, is “Should I use a heavy bat or a light bat to hit the ball farther?” And that is what is eventually addressed. In the real world, it depends on the strength and skill of the batter.

One other real world factor that is important in being able to hit the ball has to do with correct form. You want to hit the ball at some upward angle, not a flat swing. Gravity will act on the ball and pull it down, so if you hit with a flat swing, the ball will fall quickly to the ground, and you’ll hit grounders all day. Hit with an upward swing and you put the ball into a ballistic path. Then you can hit homers. (Yeah, now I figured this out. Not back when I was in little league and couldn’t hit the ball to save my life.)

My nitpick with the article is the line about needing forearms like Popeye, thus explaining Mark McGwire’s biceps. The biceps are on the upper arm, not forearm.

6

Irishman correctly points out that the problem stated constant torque. The consequence of this is that

  1. the kinetic energy of the bat at the moment it strikes the ball is independent of the mass of the bat!!!
  2. however, the larger the bat mass the larger the momentum of the bat.
    (these results are proved below)

But this still does not quite answer the question.
In a collision with the ball kinetic energy is approximately conserved (due to heating of the bat and ball) and momentum is strictly conserved. To find a solution to these equations we have to take into account the momentum of the incoming ball as thrown by the pitcher.
So far everyone has been ignoring the initial momentum/kinetic energy of the incoming ball thrown by the pitcher. Is this negligble? A simple example suffices to convince me that this may be quite significant: given a takeoff angle, the distance a ball travels is soley a function of its kinetic energy. Hence distance traveled can be used to compare kinetic energies. Fielders position themselves at a range roughly where baseballs land. Most fielders can, if pressed, hurl a ball to home plate from the field–though they rarely do in practice for technical reasons. But the point is that this equality of distance by throwing arm and batting means that the kinetic energy provided by a pitcher and that of the bat are roughly comparble–the bat of a homerun hitter might provide somewhat more energy, but it does not greatly DOMINATE the energy provided by the pitcher. this will affect the optimal bat size. I havent worked the math any farther, but clearly the issue is more subtle than so far examined.

Now setting that aside let us return to irishmans comment about constant torque. For the purposes of analysis it makes no difference if we look at this in terms of constant torque or constant force accelerating the bat, and force is a little easier to think about.

if a bat of mass m is accelerated (swung) over a distance d with a constant force f then the time to travel this distance is:
t= sqrt(2md/f)
and the final velocity is
v = ft/m = sqrt(2d*f/m)

and so the momentum and kinetic energy of the bat at the momentum of contact is:

momentum=mv = sqrt(2df*m)

kinetic energy = 1/2mv^2 = d*f

THE MASS DROPS OUT of the kinetic energy!!! thus the kinetic energy of the bat is identical regardless of the mass of the bat.

7

Interesting comment about the energy provided by the pitcher (i.e. pitch speed) being important. I think the reason we have neglected it has to do with the phrasing of the question, which is asking about comparing bat weights. With this in mind, we must assume identical pitches because the question was not about whether you could hit the ball farther off a fastball or a slow lob. Obviously case-by-case situations will have other factors than the bat size. Talking about arm size is relevant because it affects each user’s ability to use the bat. But pitches if assumed constant will not affect a change between batters.

8

Well obviously the pitch speed does matter, atleast in the extremes. For example, if a fast pitch carries significnatly more momentum than a light bat then the batter cannot hit the ball in the forward direction! In general faster the pitch the more timportant it is to have a heavier bat.

On a different tack however, its interesting to note that what babe ruth gained and lossed when he downsized his bat weight. if he went to a batsize of half the weight then his swing would be 1.4 times faster. That is the bat spends about 30% less time in the strike zone, and thus decreases the probaility of a hit. And if he does get a hit its ultimate range is less bracuse the bat is lighter. What he gained in return was a few hunderd milliseconds of delay before he had to swing the bat. For example if it took him a quarter of a second to swing the heavy bat, he could delay his swing by about 75 thousands of a seconds.

So for the down sizing to be profitable that 75 milliseconds longer of staring at the incoming ball must improve his aim and timing enough to overcome the loss of power and less tolerance of timing mismatches in the strike zone.

9

I just thought you guys missed an obvious point: the corked bat. If a heavier bat could hit a ball farther, why do some professional ballplayers risk suspension by corking their bat? A corked bat is only 1-1.5 ounces lighter (depending on the corker) but it also brings the bat’s center of gravity closer to the batter, which comes into the torque argument.
For all you Albert Belle fans out there, just ask him which he prefers.

10

Using bigkahuna’s equation for velocity with M = mass of bat,

Vbat = ft/M = sqrt(2d*f/M)

and the formula for elastic collisions (from Halliday and Resnick),

Vball_final = 2MVbat / (M + m) + Vball_init*(m-M)/(M+m)

where m = mass of ball, and Vball_init is negative, I solved for Vball_final as a function of M:

Vball_final = (2Csqrt(M) - (M-m) * Vball_init) / (M+m)

with C = sqrt(2df). I plotted this as a function of M for a few choices of C, and it does go through a maximum. You can also see this from the equations by expanding in small powers of 1/sqrt(M) for large M:

Vball_final = 2C(1-m/M)/sqrt(M) - Vball_init * (1 - 2m/M)
= -Vball_init + 2 * C / sqrt(M) + 2m/M * Vball_init

since the 1/sqrt(M) term dominates the 1/M term, for some mass of bat, the ball travels farther than in the limiting case of a very heavy bat (which is just -Vball_init). So there is an optimum bat mass.

This assumes that the distance of the swing is constant, which seems appropriate. If instead, the duration of the swing is constant, then

Vball_final = (2ft - (M-m) * Vball_init) / (M+m)

or approximately (for a heavy bat)

Vball_final = -Vball_final + (2ft + 2Vball_initm)/M

and the answer depends on whether ft + Vball_initm is positive or negative.

Either way, the answer isn’t as simple as “a baseball can be hit farthest with a heavy bat”.

11

Taking up zenBeams formula for Vball and finding the value of bat Mass that maximizes I get:

M = (Vball_init *m/c)^2

which has the nice interpreation of

M = m * ( KEball_init/ KEbat_init)

Where KE means kinetic energy.

In plain english the optimum mass of the bat for hitting the ball the furthest occurs when the it is equal to the mass of the ball times the ratio of the initial kinetic energies of the ball and the bat.
(keep in mind that the assumption of constant torque means the bat’s energy is fixed, independent of its mass).

Now that is cool. Anyone know how much a bat and ball weigh ?

(regarding corking: corking makes the collision ineleastic so that there less recoil from the bat, and hence more momentum is delivered to the ball. I believe this can up to double the ball speed)

12

Thinking about the above result:
If a batter becomes stronger, and hence his kinetic energy goes up, then he wants to use a lighter bat to hit the ball the furthest.
If a batter expects fast balls, then he wants to use a heavier bat than if he expects slow balls. Conversely, once he has entered the batters box with a given weight bat, say a heavy one, he might choose to pass up slow pitches and wait for a fast ball that was optimized for his bat weight.

13

How does that make the collision inelastic? Do I misunderstand “corking” the bat? I thought it was filling a hollowed out section of the bat with cork, thereby making it lighter. Not so?

14

then
M * KEbat_init = m * KEball_init
so
M * (M Vbat^2/2) = m * (m Vball^2/2)
or
(M * Vbat)^2 = (m*Vball)^2

So for the bat with optimal weight, the bat and ball have equal (and opposite) momentum at collision.

15

I see the math but i think it has to be wrong. If it were true then the bat would recoil at the same velocity it was swung with. This is empirically baloney. ergo: math error somewhere.

16

I’m speculating here but I think there are two possible reasons to cork a bat. One would be to alter its moment of inertia without altering its mass. I dont think this is the main issue. The second and more plausible reason is to make it work like a dead blow hammer. A dead blow hammer is a hammer head filled with lead shot which makes the blow in elastic. When an inelastic hammer strikes a surface or nail it does not recoil. Since momentum is strictly conserved then all of the momentum gets transfered to the nail.

But I havent really put on my thinking cap here so maybe i’m blowing smoke on this corking issue

17

Yeah, i think you’ve got the wrong grasp on corking. Corking a bat involves drilling a hole into the top of the bat and hollowing out a section about six to eight inches deep and half to an inch in diameter. This section is then filled with cork to keep the bat from making a “hollow” sound when it’s used. I imagine some people have tinkered with ways to make the bat like a dead blow hammer, but corking seems to be the perfect crime in baseball.

18

Bigkahuna wrote:

“The second and more plausible reason is to make it work like a dead blow hammer. A dead blow hammer is a hammer head filled with lead shot which makes the blow in elastic. When an inelastic hammer strikes a surface or nail it does not recoil. Since momentum is strictly conserved then all of the momentum gets transfered to the nail.”

IMHO the advantage of corking may be that a bat becomes MORE elastic. A more loosely strung tennis racket provides more power and less control than a more tightly strung racket. Greater elasticity increases the amount of time that the bat (or tennis racket) is in contact with the ball, thereby increasing the ball’s momentum.

19

I just wanted to add a couple of comments from what I’ve read. First of all, Force does not equal mass times acceleration. The sum of forces equals mass times acceleration. There is a big difference. Secondly, the question posted is a question of momentum and impact. Since momentum is equal to mass times velocity, if you can increase either variable you will increase your momentum or, in this case, the speed of the ball from the bat.

20

A loosely strung tennis racket hits a ball harder than a tightly strung racket because with a loosely strung racket, the ball stays on the racket longer. This creates kind of a trampoline effect, but in physics terms, the impulse of the force that the racket exerts on the ball is increased because the amount of time the ball stays in contact with the racket increases. I don’t see how corking could do this to a bat, but then again, I’ve never personally hit with corked bat.