Roller/Wheel Ball Shooter Physics
 

With following post I am trying to summarize physics of a wheel based shooter. (e.g. https://youtu.be/Kd1FaSNoDiM?list=PL...JScoQterdV 4). I appreciate any corrections.

Assumptions:

• Ball kinetic energy is composed of rotational energy and translational energy. http://hyperphysics.phy-astr.gsu.edu/hbase/rke.html
  
• After launch the ball’s kinetic energy came from the rotational energy of the rollers and the electric energy of the motor during the short contact time between the ball and the roller is negligible (it takes seconds to spin up the rollers and it takes fractions of a second for the ball to pass the shooter).
  
• The amount of transferred energy depends on the friction between the ball and the rollers. The friction is defined by the normal force times the friction coefficient. The normal force is the ball compression. The energy is defined by the friction times the distance the object is pulled with that friction, which is the short distance along the roller circumference when the ball is in contact with the roller.
  
• The ratio between rotational and translational energy is defined by the speed of the contact points of the ball at launch. If one side is stationary, and the other side is the roller, then we have a ball that spins.

We want maximum launch speed of the ball. Maximal launch speed is desired because aiming a fast projectile is easier than a slow projectile (gun versus bow). To achieve this we can optimize the following:

a) reduce spin of the ball so that all kinetic energy is translational and none rotational. This requires two rollers.

b) increase rotational energy stored in the rollers:

b1) larger inertia in the rollers (goes linear)

• heavier roller wheels
• larger roller wheels
• more weight on the outside of the rollers
• more roller wheels (goes linear)
• more rollers (goes linear)

b2) faster rotation of the rollers (goes by omega square). Fast rotation is achieved with different gear ratio not with more motors, more motors will achieve final angular velocity faster

c) increase friction between ball and rollers (goes linear):

• increase normal force through more compression
  
• increase friction coefficient through sticky coating on rollers or teeth on rollers

d) increase the contact time of the ball with the rollers:

• larger wheel diameter (goes linear, larger roller gives in addition also more intertia)
• multiple rollers in parallel or in series (goes linear)

e) reduce inelastic component of compression. this can not be controlled, but less compression gives less inelastic loss, but that gives also less friction, so this needs to be optimized. There is inelastic component because the ball is made of foam and the air moving through the pores causes friction. Inelastic component can not be regained and is lost. If you compress a spring, almost all energy is regained again when the spring extends.