# Muzzle velocity given test range, time of flight and ballistic properties.

by Carpet_Diver
Tags: ballistic, flight, muzzle, properties, range, test, time, velocity
 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 $\large \dot{x}_0 = \frac{r}{T}$ 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, $\large X = T \sqrt{(\frac{r}{T})^{2}+(\frac{g T}{2})^{2}}$, 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 $\Large \ddot{x} = \frac{C \dot{x}^{2}}{M}$ $\dot{x}_{0}$, 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; $\LARGE \dot{x}(t) = \frac{\dot{x}_{0}}{1 + \frac{t C \dot{x}_{0}}{M}}$ and an expression for distance, as a function of time; $\LARGE x(t) = \frac{M}{C} ln(1 + \frac{t C \dot{x}_{0}}{M})$ I am not really sure what to do next, to get what I want; an expression for $\dot{x}_{0}$ as a function of X, T, C and M.
 Quote by Carpet_Diver 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 $\large \dot{x}_0 = \frac{r}{T}$ 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, $\large X = T \sqrt{(\frac{r}{T})^{2}+(\frac{g T}{2})^{2}}$, 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 $\Large \ddot{x} = \frac{C \dot{x}^{2}}{M}$ $\dot{x}_{0}$, 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; $\LARGE \dot{x}(t) = \frac{\dot{x}_{0}}{1 + \frac{t C \dot{x}_{0}}{M}}$ and an expression for distance, as a function of time; $\LARGE x(t) = \frac{M}{C} ln(1 + \frac{t C \dot{x}_{0}}{M})$ I am not really sure what to do next, to get what I want; an expression for $\dot{x}_{0}$ as a function of X, T, C and M.