Posted by: Mike Myers () on Wed Oct 23 11:17:26 2002

I was just wondering if there were anyone here who could tell me how to figure, given the velocity of the bat and the velocity of the ball, the velocity of the ball leaving the bat. Complex physics welcome :)

Jacob,

Consider an object’s momentum is its mass x its velocity. Energy delivered by the body during swing will cause momentum in the bat. Since the body is continuously supplying energy throughout the swing, bat momentum is increasing until time of impact.

Conservation of momentum states that net momentum of a system before a collision is identical to net momentum of a system after a collision. Thus, if you have 1 billiard ball travelling toward another billiard ball, each will have its own mass and velocity. Net momentum before the collision is (m1v1 + m2v2). After impact, the masses are unchanged, however, final velocities will differ as the balls separate. Net momentum is now (m1v1f + m2v2f). Conservation of mometum states: m1v1 + m2v2 = m1v1f + m2v2f.

In baseball, just before the bat collides with the ball, system momentum is: mbat x vbat + mball x vball. Just after the collision, the bat will recoil, and the ball will fly. Final system momentum is: mbat x vbatf + mball x vballf.

Equating initial and final state momentum:

mbat x vbat + mball x vball = mbat x vbatf + mball x vballf.

Mass of bat and ball do not change – only their relative velocities. Now, the official 2000 rules (www.mlb.com) state the ball must weigh between 5 and 5.25 ounces. There is no regulation on the mass of the bat, only a length and diameter restriction. Let us assume the bat is a typical 30 oz. Then mbat is approx 6 x mball. Therefore:

6 x mball x vbat + mball x vball = 6 x mball x vbatf + mball x vballf
or mball (6vbat + vball) = mball (6vbatf + vballf)
6vbat + vball = 6vbatf + vballf

With all constants accounted for, the only factors which remain to affect ball launch speed after contact, are pitch speed (vball) and batspeed (vbat) just before contact. Also, one can see bat speed has a 6 x larger factor on system momentum than ball speed, so developing high bat speed is critical in developing home run power. If you increase your bat speed by just 5mph at contact, this will have the same effect as contacting a pitch thrown 30mph harder.

The human body can produce bat motion with a variety of methods. The question is, with fixed energy, how is this motion most efficient to produce maximum batspeed? The answer is to realize we are only concerned with maximizing speed in the part of the bat which contacts the ball (ideally, the sweet spot).

Thus, consider the bat to contain many small sections, with a small mass in each section. Each with separate motion characteristics. Now, each small bat section will have its own ((small m) x v) characteristic at every instance in time. Since we want the sweet spot of the bat to have highest possible velocity, then we must realize, for a bat with fixed energy input, higher speed in some bat parts, must mean lower speed in other bat parts.

What is the only possible bat motion which sacrifices speed maximally in certain parts, so as to maximize speed at the sweet spot? The answer is: rotation about the knob. Actually, the far end of the bat will be travelling even faster than the sweet spot, but we cannot optimize our mechanic any further since the bat is a rigid body which must maintain its shape throughout movement.

Now, what if the human body cannot harness all its energy into keeping the knob on a stationary axis and accelerating the bat head maximally? We must re-examine the optimal energy-to-bat transfer mechanic (rotation about knob) to consider the next best alternative. The body has long and massive limbs whose chief power sources operate at a distance, causing arm and leg rotation about joints far from the bat. We realize the human body itself, should ideally rotate as well to produce maximal torque (force of rotation) on the hands and consequently, the bat. Thus, the bats’ ideal axis of rotation to produce maximal bat speed, should actually lie about the center of the rotating human body.

Regards,
Mike.