A Performance Analysis System for the Sport of Bowling

known, the linear momentum (and thus the linear velocity) for that revolution can also be
found. Once the linear velocity for each revolution is known, the location of each
revolution of the ball and the ball loft distance, relative to the foul line, can be found.
Putting those assumptions and deductions into more formal terms, the assumption that the
ball does not lose energy over time means that the ball possesses constant energy
throughout its trip to the pins, and that its energy at any instant is equal to its energy at
any other instant [8].
The total energy of the ball (E) is the sum of its potential (P) and kinetic (K) energies:
E = P + K.
Since the ball rolls on a flat, level lane surface, there is no potential energy (P = 0), and
E = K.
Since it is assumed that the ball has constant energy from its release at the foul line to its
impact with the pins,
K R = K i = K P ,
where K i is the energy of the ball for each revolution i between release (K R ) and pin
impact (K P ).
The assumption of constant energy implies that energy losses due to vibration, heat, wind
resistance, and noise generation are negligible. Therefore the only components of the
energy of the ball are its angular momentum (K ω ) and its linear momentum (K v ), thus
K = K ω + K v .
The angular momentum of an object with angular velocity ω is given by
K ω = ½Iω 2 ,
where I is the moment of inertia of the rotating object.
The moment of inertia I of a bowling ball with mass m and radius R falls somewhere
between the moments of inertia of a solid sphere and a spherical shell, such that
2 / 5 mR 2 < I < 2 / 3 mR 2 .
The actual moment of inertia for a given ball will fall between these two values, but
closer to the first, since the density of a bowling ball in the 12-16 pound range is fairly
constant across the radius of the ball [8].
The 2 / 5 R 2 term is a constant, where R is the radius of the ball (nominally 4.3"), so if
k = 2 / 5 R 2 = 0.4 · (0.3583 ft) 2 = 0.05136 ft 2 ,
then
K ω = ½kmω 2 .
The linear momentum of an object with mass m and linear velocity v is
K v = ½mv 2 .
Therefore, the total kinetic energy K of the ball, under the assumptions, is given by
K = ½mv 2 + ½kmω 2
= ½m(v 2 + kω 2 ), where k = 0.05136 ft 2 .
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