Physics Behind Falling Bullets

Many a times, in films, police shoot bullets up in the air to grab the attention of the people around. I used to wonder, what happens to the bullet? Well today I am writing this post to explain the same.

If we consider the height of the policeman who shoots the bullet to be negligible and also the air resistance to be very little to be considerable, then the maximum height reached by the bullet is given by
where u is the initial velocity of the bullet and g is acceleration due to gravity.
At the highest point the velocity of the bullet becomes zero for a very small instant of time. The final velocity of the bullet just before hitting the ground would be
by replacing s with u²/2g we get 
This tells us that the magnitude of velocity with which the bullet will strike the ground is equal to the magnitude of velocity of projection. The other way of proving this is the kinetic energy of the bullet will be converted into its potential energy. From the highest point, when the bullet will begin to move, its potential energy will  again be converted into kinetic energy. Since there is no loss of energy, and as the mass remains constant, we can say that the magnitudes of final and initial velocities must be the same. But wait. This is only under ideal conditions when the air resistance is negligible. What happens when we take into account the air resistance as well?

When a body is projected up, it won't reach the predicted maximum height due to the air resistance. When a body falls down from a certain height, due to g its velocity keeps on increasing. As the velocity increases, the air resistance also increases. Down the path (provided that there is sufficient time from the time of projection to the time of landing), there comes a point when the magnitude of air resistance equals the magnitude of gravitational pull (weight). Since they act in the opposite directions, they cancel out each other and the net force acting on the body becomes zero. This also means that the acceleration of the body is zero (Newton's First Law of Motion). The body then moves with a constant velocity. This is the maximum velocity that can be achieved by the body during its fall. Hence this is called terminal velocity.
When a bullet approaches its maximum height, as its mass is less and the velocity keeps on decreasing, wind can blow it away easily. This changes its path and it becomes highly unlike that it would hit a person but it is not entirely impossible.For example t
he terminal velocity of a .30 caliber bullet is about 880 m/s, which is highly lethal. 
Next time before shooting a bullet, think about the physics behind its motion!

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