To calculate the draw length required to achieve a specific initial velocity with a recurve bow, we need to consider the draw weight, arrow weight, and the energy stored in the bow.
First, we need to convert the 9 Newton weight applied at 1cm draw back into joules of energy. The formula to calculate energy is:
Energy (Joules) = Force (Newtons) x Distance (meters)
Since the draw back is 1cm (0.01m) and the weight applied is 9 Newtons:
Energy = 9 N x 0.01 m = 0.09 Joules
Next, we can calculate the energy stored in the bow using the equation:
Energy (Joules) = (1/2) x Draw Weight (Newtons) x Draw Length (meters)
We need to rearrange this equation to solve for Draw Length:
Draw Length (meters) = (2 x Energy) / Draw Weight
Given that the arrow weight is 0.027kg and the desired initial velocity is 13.48 m/s, we can calculate the momentum of the arrow:
Momentum (kg·m/s) = Arrow Weight (kg) x Initial Velocity (m/s)
Momentum = 0.027 kg x 13.48 m/s = 0.36396 kg·m/s
Now we can calculate the Draw Weight:
Draw Weight (Newtons) = Momentum / Draw Length
Draw Weight = 0.36396 kg·m/s / Draw Length
Finally, we can substitute the values into the equation to find the Draw Length:
Draw Length (meters) = (2 x 0.09 Joules) / (0.36396 kg·m/s)
Draw Length ≈ 0.4934 meters (or approximately 49.34 centimeters)
Therefore, to achieve an initial velocity of 13.48 m/s with an arrow weight of 0.027kg and a 9 Newton weight applied at 1cm draw back, a draw length of approximately 49.34 centimeters is required.