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Catapults

Galileo experimented with the acceleration of an object starting at rest and realized that the velocity of balls that roll down inclined planes and falling objects increase by the same acceleration.  He learned that the final velocity can be calculated by multiplying the acceleration and the elapsed time.  Galileo's theory of gravity is what brings all objects back towards the ground, and was the basis for Newton's laws of motion.  An example of the constant pull of gravity is, if a person were to shoot a gun horizontally and drop a bullet from the same height as the barrel, both the fired bullet and the dropped bullet would reach the ground at the same time.

Galileo made several formulas to calculate different variables of a projectile:
From Rest Formulas:
Vf = aΔt
Δd= ½aΔt^2
Vf= (2aΔd)^½

 With Initial Velocity
Vf = Vo + aΔt
Δd = VoΔt + ½ aΔt^2
Vf = (Vo 2 + 2aΔd)^½

An object launched from a catapult uses the equations associated with initial velocity.

An example of a catapult shot that possibly could've been used in medieval ages is when the projectile was launched at 40° with a projected initial velocity of 22m/s.  Before any variables of the shot can be found, the x and y components of the initial velocity must be calculated.

 For the y component:

 Voy = Vo Sinθ
      = 22(Sin 40)
      = 14.1 m/s

 For the x component:

 Vox = Vo Cosθ
       = 22(Cos 40)
       =16.9 m/s

Before calculating the distance, the known y values need to be used to find the time.  The equation, Δdy =VoyΔt + ½ aΔt^2 will be used. 

Δdy = VoyΔt + ½ aΔt^2
0 = Δt(Voy + ½ aΔt)
Δt= -[(2)(14.1)]/-9.8
Δt=2.88s

Now that the time is calculated, the known x values and the time will be used to calculate the total horizontal distance of the projectile.  The equation Δdx = VoxΔt is needed to solve for the distance.

Δdx = VoxΔt
Δdx = (16.9)(2.88)
Δdx = 48.67m





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