Bungee Ballista Part 3 PDF Print E-mail
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Tuesday, 27 October 2009 03:14

Bungee Ballista Part 3 - Physics


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Calculating the Maximum Possible Muzzle Velocity

 The amount of energy stored in the spring is equal to the amount of work you did when you pulled the projectile backwards.   In physics, amount of work done is equal to Force*Distance.   So increasing either
the spring force, or the distance you pull back against the spring will result in a faster projectile.
 Work = Force * Distance  =  Total Energy Stored in Bungee when Stretched
 So ideally, after the projectile has been accelerated by the bungee and exits the front of the ballista, most of this energy should now be in the projectile in the form of Kinetic Energy.    Something with mass which is moving has a kinetic energy equal to:
½ MV2  = .5 * mass * velocity * velocity
  Kinetic Energy is the amount of energy it takes to get something up to speed.  It is also the amount of energy which will be released when it is stopped.   By the time the projectile comes to a rest, all of this energy will have be transferred to another form.  Usually this would be heat, or perhaps some work may be done by smashing whatever it hits, or if it hits something that moves, then whatever it hits will have inherited some kinetic energy.
  So taking our two equations, we set the kinetic energy of the projectile equal to the work done in stretching the bungee:
½ MV2  = Force  * Distance
  Where M is the mass of the projectile
  V is the muzzle velocity of the ballista
 Force is the average force it takes to pull back against the bungee
 Distance is the how far the shuttle gets pulled back against the bungee.
Our equation ignores all the factors which waste our energy, such as energy lost in bungee cord because they are not ideal springs, friction against the track, and energy wasted accelerating the shuttle, the ropes, the wheels, and the bungee itself.   For light projectiles, these effects can be huge, but for heavy projects, less so.
This equation can be rearranged to solve for V
V=   sqrt( 2 * F * D / M )
 This puts an upper limit on the final velocity of the projectile.
 The Bold Inventions Ballista uses bungee cord for a spring.    A typical bungee cord will stretch to more than twice its original length.   At that point, the cord suddenly becomes much harder to stretch further, and if you continue, you will break it.

 The maximum load on one bungee cord is typically around 10 or 20 pounds.  A handy tool to have is a luggage scale, which have become quite common now that the airlines like to charge passengers large fees for bringing overweight luggage on an airplane.
  In this design, 60 feet of bungee cord was purchased from Campmor.com.  The springs were made by bundling three cords together on each side, so the average total spring force is approximately 15 * 6 = 90 pounds.
  Note that the tension is not constant.  It increases as the cord is stretched.  For the purposes of this estimation, we choose an 'average' tension.   A more accurate way to calculate the energy would be to measure the tension at regular intervals, and then integrate.
  We will work in metric units.
90 pounds = 400 Newtons
7 feet = 2.13 meters
8 pound projectile has a mass of  3.6 kilograms
So a maximum value for our muzzle velocity would be
sqrt( 2 * 400 * 2.13 / 3.6 ) = sqrt(  473 ) = 21.75 meters/sec
And converting back to miles per hour:   21.75 meters/sec = 48 miles per hour
An 8 pound projectile moving at 48 miles per hour is lethal, like a rock dropped from an expressway bridge.   A ballista is a very dangerous thing.
Calculating the Maximum Possible Range
   Wikipedia has a good article on trajectories.   
   The maximum range of the projectile while ignoring the effects of wind resistance is given as
Max possible range = v2 / g
   This occurs when the ballista is launching the project at a 45 degree angle.
R = (21.75 * 21.75) / 9.8 =   48 meters  = 157 feet.
  So this means if you have neighbors within 300 feet of your back yard, it probably isn't the best place to test fire this ballista.   
Last Updated on Thursday, 29 October 2009 02:17

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