Ballista Physics
Jason Hoisington
Phys 212
    The design of the ballista is such that the force applied from the projectile comes from the tension of the twisted ropes.  The ropes, when the tension is released, tend to return to their rest state with minimum tension, much like a spring would expand after being compressed.  Using this similarity, the assumption can be made that the forces act in a similar way, and that Hooke's Law can be applied to give at least a general idea of the nature of the force applied by the ballista arms.  Doing so, we get
where F is the force applied, d is the distance that the arms are drawn back, and k is the force constant of the ropes when under tension.  Using this relationship assumes that the force constant is constant, or that moving the arms back 2 meters gives twice the force that moving them back 1 meter would do, which is most likely not correct, but close enough for a general assumption of the force to be made.  When the force is applied, the projectile is accelerated to the end of the ballista, at which point it released with a velocity v and an angle q from the horizontal.  The velocity can be found using the kinematic equation
where a is the acceleration and x is the length of the ballista that the projectile is accelerated upon.  Since F=ma and F in this case is -kd, the equation can be simplified into
After it is released, the projectile obeys the laws of projectile motion, disregarding air resistance.  Therefore, the range of the ballista can be given by the equation
where g is the accleration due to gravity.
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