Create a Model of a Moving bullet from an M-16 rifle. State any assumptions on which your model is based. Two cases arise for a moving bullet.

i)- Moving Horizontally

ii)- Moving Vertically

i)-

When we look at horizontal motion and completely ignore vertical motion. This way we won’t have to worry about the angle that the gun is shot at and stuff like that. Once the bullet leaves the barrel, I will have the following (modified) free body diagram:

Free Body Diagram

In this one-dimensional model , there is only one force acting on the bullet – the force of air resistance. Note that this force is in the opposite direction as the velocity . There is a fairly simple model for the air resistance force that says:

In this model, rho is the density of air, A is the cross sectional area of the bullet, C is the drag coefficient for the object (depends on the shape). v-hat” is a vector in the direction of the velocity, but with magnitude 1 and no units. So, to sum up, the air resistance force is in the opposite direction as the motion and proportional to the square of the magnitude of the velocity. The faster it goes, the greater the air resistance force.

Now, this should be easy to apply the momentum principle or work energy theorem: So, momentum principle says,

A net force on an object changes its momentum, where momentum is the product of mass times its velocity. (velocity, and therefore momentum, is a vector quantity)

And

But we will prefer momentum principle only. So,

Where is change in momentum of bullet with respect to time. This is actually a problem because the force is not constant. If there is a force on an object, it’s velocity will change. Since the force depends on the magnitude of the velocity, it will change also. ii)- After the bullet leaves the gun, it has forces acting on it like this: Free Body Diagram

As bullet goes really fast. So , It is not safe to assume that the drag coefficient (C) is constant with speed. Thus at terminal velocity Net Force = weight + air resistance So,

And as we know that for momentum

Thus net change in momentum is given by