newtonian mechanics - Deceleration of a bullet

Taking into account gravity, air resistance, and wind velocity and direction. (And also temperature if it's actually relevant.) I know the muzzle velocity, the total distance the bullet traveled, its diameter, mass, length, and the angle from which it was fired at. Now, I do not know its drag coefficient, so how do I calculate it? (The muzzle velocity is supersonic.)

The question is, what is the velocity of the bullet by the time it traveled the total distance? What are the formulae that I would need to use to calculate that?

Much obliged.

Answer

Let's start off by writing down all the forces the bullet experiences from the moment it leaves the muzzle all the way to the moment it lands on the ground:

1. Gravity:

$\vec{F}_g=- m g \hat{y}$

where $\hat{y}$ is a unit vector pointing upwards, and that's why there's a minus sign there, to make it point downwards.

2. Drag force:

$\vec{F}_d =- \frac{1}{2} \rho v^2 C_D A \hat{v}$

which is in the opposite direction of the bullet velocity. $\rho$ is the density of air, $v$ is the speed of the bullet, $A$ is the cross section of the bullet, and $C_D$ is the drag coefficient. Similarly, $\hat{v}$ is a unit vector pointing in the direction of the velocity of the bullet.

Now that you have all the forces, you just need to plug them in Newton's equation of motion to calculate the trajectory of the bullet:

$\vec{F}_{tot} = m \vec{a}$

where $\vec{F}_{tot} = \vec{F}_g + \vec{F}_d$, and $\vec{a}$ is the acceleration vector. $m$ is the mass of the bullet.

Now all you need to do is solve this differential equation. As far as I know this differential equation cannot be analytically solved (edit: well, as @Gert has mentioned in a comment below, this is not true. Reference. Nonetheless, the rest of this answer is still accurate as a numerical approach), therefore, I would use the shooting method as the simplest approach. I would start with a drag coefficient close to that of the air at let's say 25 degrees Celsius, and calculate where the bullet lands. If it was further than the given distance, I would increase the drag coefficient and if it was shorter, I would decrease the drag coefficient. You can keep doing this until you get the desired accuracy.