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Muzzle velocity given test range, time of flight and ballistic properties. 
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#1
May912, 03:27 PM

P: 2

This isn't really homework, but I figure this is the most appropriate place to post this... Please do say if you think there is a more likely place it will get answered  I'm new!
The problem statement, all variables and given/known data I am trying to calculate the muzzle velocity of an air rifle. I can find the time of flight over a known horizontal range. I know the mass and quadratic velocity damping constant due to air drag of the pellet. I also know that over a reasonable range (20m), air drag cannot be ignored, so [itex]\large \dot{x}_0 = \frac{r}{T}[/itex] is not true. I will assume the trajectory of the pellet is a parabola, because the trajectory is so flat a parabola is a good enough approximation to real life. I will treat the motion decay problem as one dimensional (along the parabola arc length), so gravity can be ignored from here on. Relevant equations X, parabola arc length is known, [itex]\large X = T \sqrt{(\frac{r}{T})^{2}+(\frac{g T}{2})^{2}}[/itex], where r is horizontal range, and g is gravity. T, time of flight is known C, quadratic velocity damping constant is known M, pellet mass is known Acceleration due to quadratic velocity damping is given by [itex]\Large \ddot{x} = \frac{C \dot{x}^{2}}{M}[/itex] [itex]\dot{x}_{0}[/itex], the initial velocity, is unknown. The attempt at a solution so far I have already worked out the arc length, as seen above. I have come up with the following expression for velocity as a function of time, which may, or may not be useful; [itex]\LARGE \dot{x}(t) = \frac{\dot{x}_{0}}{1 + \frac{t C \dot{x}_{0}}{M}}[/itex] and an expression for distance, as a function of time; [itex]\LARGE x(t) = \frac{M}{C} ln(1 + \frac{t C \dot{x}_{0}}{M})[/itex] I am not really sure what to do next, to get what I want; an expression for [itex]\dot{x}_{0}[/itex] as a function of X, T, C and M. 


#2
May912, 04:23 PM

P: 317




#3
May912, 04:35 PM

P: 2

Wow, thanks, I am obviously in need of sleep!



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