Monday, November 21, 2011

The Physics of the Bullet Drop Problem

A typical physics problem combines two identical bullets, one shot from a gun at a given height and the other dropped from the same height. Analyzing when they hit the ground suggests that, ideally, they hit the ground at the same time. MythBusters actually tested this problem and looked at the reality of obtaining these results.

When the bullets are both fired and dropped there are forces acting on them, the force of gravity acting in the downward direction and the force of air resistance acting in the direction opposing the motion of the bullet.

If we were to look at this problem in a vacuum the air resistance could be ignored and it would be very clear that the force due to gravity would be the same on both bullets. The force that the gun applies to the fired bullet is only acting horizontally and thus determines how far in the x-direction the bullet will move, not how long it will take. Since the height the bullets are dropped from is the same and the force causing their downward motion is the same both bullets would hit the ground at the same time.

However, as the MythBusters found out this is not exactly true. Because of air resistance acting on the bullets there is a slight difference in the time it takes for them to reach the ground, though very, very small. As discussed in class air resistance (or the drag force) is expressed as

But in order to determine this force on the fired bullet as it relates to the time that it takes for the bullet to fall we must break this force into components. Because of this breakdown into components it can be determined that the air resistance on the fired bullet actually causes it to take longer to hit the ground than the dropped bullet, also experiencing air resistance.

The MythBusters demonstration made it clear that, though the bullets won’t hit at exactly the same time the difference in the time is so small it has to be measured with a high-speed camera and using the images can be calculated to be about 39.6 milliseconds, a time that is smaller than the human eye can detect. (start at around 2:10)

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