Debunking the Tumble Theory

In 1964, a riddle circulated within the technical community. They asked, "What happens to a fast moving bullet?" Their punch line was, "Very, very little!"

The tumble theory died of a birth defect. Arlene's specter was born without an impulse. (1) According to the initial autopsy information the single bullet passed through Kennedy without striking a bone. Clearly collision between the bullet and homogeneous tissue could not induce any tumbling.

Before they could bury the tumble theory, something astonishing happened. While reviewing an autopsy x-ray, they found the single bullet struck a glancing blow to a bone. This discovery of the missing impulse resuscitated the tumble theory.

A rapid pace made life difficult for the tumble theory. First they demanded the bullet tumble one-quarter of a revolution while traveling between Kennedy and Connally. At 1600 feet per second, the single bullet had less than two-thousandths of a second to tumble. (2)

The single bullet could live while tumbling at more than 1000 radians per second. (3) However, the rapid pace of the single bullet caused complications. An initial symptom was the mere 0.00003 second for the single bullet to accelerate. (4) Shortly afterwards, the tumble theory developed hyperangular acceleration. Preliminary examination showed the tumble theory had a life threatening acceleration count of nearly 37 million radians per second per second. (5) They immediately consulted specialists.

At first the specialists were optimistic. The single bullet was small and had a minuscule moment of inertia. They hoped the tumble theory could survive a torque of 24 lb ft. (6) Their hope turned to despair as they realized the small bullet had a moment arm of one-fortieth of a foot. Grimly they calculated an accelerating force of more than 900 pounds (7) and a reaction force of nearly 1300 pounds (8) from the struck bone. As the specialists advised them to make arrangements, alarms on life monitors sounded. The vital signs of the tumble theory ebbed to dangerous levels.

Experts placed the tumble theory on life support. They performed a radical numerallectomy and the tumble theory lived happily ever after. Well almost happily, those damning numbers kept growing back.

Notes

(1) Newton equated change in motion to the impulse.

(2) Given:

Distance between Connally and Kennedy = 2.33 ft.
Speed of bullet = 1600 ft/sec
Calculate time of travel:
2.33 ft/1600 ft/sec = 0.0014 sec

(3) Given:

Bullet rotated ninety degrees (pi/2 radian)
Calculate angular speed:
1.57 radian/0.0014 sec = 1078 radian/sec

(4) Given:

Length of bullet = 0.1 ft
Speed of bullet during collision = 1700 ft/sec
Calculate duration of collision
.05 ft/1700 ft/sec = 0.000029 sec
Note: The half length of the bullet is the most favorable length to induce net rotational acceleration.

(5) Given:

Change in angular speed = 1078 radian/sec
Duration of collision = 0.000029 sec
Calculate angular acceleration
(1078 radian/sec)/(0.000029 sec) = 36,600,000 radian/sec²

(6) Given:

Moment of inertia of bullet = 0.000000651 slug ft²
Calculate accelerating torque:
(0.000000651 slug ft²)(36,600,000 radian/sec²) = 23.83 lb ft

(7) Given:

Average moment arm = 0.025 ft
Note: The moment arm varies linearly from zero to the half length of the bullet. This gives an average moment arm length of one-quarter the length of the bullet.
Calculate accelerating force:
(23.83 lb ft)/(0.025 ft) = 953 lb

(8) The most favorable circumstance to induce angular acceleration occurs when the reaction force acts at a 45-degree angle to the moment arm. Under the best conditions the reaction force becomes 1,348 lb.