# Galileo's Superposition and Bullet Trajectories

Q:
I am deployed with a group of soldiers and one of them was talking about an epsiode of myth busters. According to the soldier if you fire a projectile (bullet) from a rifle and that projectile does not rise in its trajectory. The fired bullet and a bullet dropped at the same time will hit at the ground at same time. Is this true and if so why doesn't the energy from firing the bullet from the gun increase its time in the air? Thank you, Joe Kennedy
- Joe Kennedy (age 49)
Afghanistan
A:
Galileo discovered that you could treat the trajectories by treating the vertical motion and horizontal motion separately. Newton provided a nice language to discuss the reasons.

Let's temporarily imagine there's no air friction. The only force on the bullet is the vertical gravity. The downward acceleration is the same for the horizontally shot bullet and for the simply dropped bullet. So they hit the ground at the same time. The energy that went into the bullet doesn't prolong the time it spends in flight unless it is aimed somewhat upwards.

In the real world there's air friction. That causes a force opposite to the bullet's velocity with respect to the air. If this force were simply proportional to the velocity, you could still treat the vertical and horizontal velocities separately. Then the falling bullet and the horizontally fired one would take the same time to hit. The actual air friction is more nearly proportional to the square of the velocity. That means that as the bullet starts to fall, there will be a bit more upward air friction force on the fast-moving bullet. So I believe that the horizontally fired bullet will take slightly longer to land. The simpler same-time answer is an approximation that works when you can neglect the air friction.

Mike W.

(published on 09/29/2012)

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