Mathematical Explanation of The TREB Physics
There are several variables to consider when calculating the physics of a pumpkin launch through the air by a trebuchet. To calculate the path of a pumpkin you must know the angle of release, the initial velocity at release, and the mass of the pumpkin. You can calculate the maximum height using equation 1
Equation 1: Max height=((initial velocity)2sin2(angle of release))/2(gravitational constant)
Now not considering the minimal effect of air resistance on the pumpkin we are able to calculate the range of its flight. Using this second equation
Equation 2: Range=initial velocity/gravitational constant
So from these two equations we can find the distance our pumpkin traveled, and the highest point of its path.
Conversely we can use equation 2 to calculate the velocity using the distance fired, and then use equation 2 to calculate the angle of release. Using this information we can get an accurate picture of what happened and use this information to refine our trebuchets.
Next we can use this information to calculate the energy involved throughout the process. The Potential energy is simply the total amount of energy that you counter weight can produce, based on how high it is off the ground.
Equation 3: PE=Mass*Height*Gravitational constant
Using equation 3 we find the PE to be of our tebutche to be 1744.22J. Using this we can calculate the ideal velocity the pumpkin when it leaves the pouch.
Equation 4: IV=(2*PE/Mass)
From equation 4 we find that the ideal velocity is 39.82m/s. Unfortunately a lot energy is lost to friction, air resistance, and if the pumkin leaves the pouch before the trebuchet stops moving then all that extra movement is wasted energy. So based on the actual distance the trebuchet fired and the velocity it was released at we can find the kinetic energy at the point of release.
Equation 5: KE=(Mass*(Initial velocity)2 )/2
From this we find the Kinetic energy to be 396.06J which is significantly less than the potential energy. Now we can better understand the quality of our trebuchet by calculating the efficiency.
Equation 6:Efficiency=Initial velocityy2/Ideal velocityy2
Now we can calculate our efficiency to be 22.71%. So even though our trebuchet did well there could have been major improvements to increase the efficiency.
Equation 1: Max height=((initial velocity)2sin2(angle of release))/2(gravitational constant)
Now not considering the minimal effect of air resistance on the pumpkin we are able to calculate the range of its flight. Using this second equation
Equation 2: Range=initial velocity/gravitational constant
So from these two equations we can find the distance our pumpkin traveled, and the highest point of its path.
Conversely we can use equation 2 to calculate the velocity using the distance fired, and then use equation 2 to calculate the angle of release. Using this information we can get an accurate picture of what happened and use this information to refine our trebuchets.
Next we can use this information to calculate the energy involved throughout the process. The Potential energy is simply the total amount of energy that you counter weight can produce, based on how high it is off the ground.
Equation 3: PE=Mass*Height*Gravitational constant
Using equation 3 we find the PE to be of our tebutche to be 1744.22J. Using this we can calculate the ideal velocity the pumpkin when it leaves the pouch.
Equation 4: IV=(2*PE/Mass)
From equation 4 we find that the ideal velocity is 39.82m/s. Unfortunately a lot energy is lost to friction, air resistance, and if the pumkin leaves the pouch before the trebuchet stops moving then all that extra movement is wasted energy. So based on the actual distance the trebuchet fired and the velocity it was released at we can find the kinetic energy at the point of release.
Equation 5: KE=(Mass*(Initial velocity)2 )/2
From this we find the Kinetic energy to be 396.06J which is significantly less than the potential energy. Now we can better understand the quality of our trebuchet by calculating the efficiency.
Equation 6:Efficiency=Initial velocityy2/Ideal velocityy2
Now we can calculate our efficiency to be 22.71%. So even though our trebuchet did well there could have been major improvements to increase the efficiency.