Can an Arrow Fired Straight Up Fall Fast Enough to Kill You
1Can an Arrow Fired Straight Up Fall Fast Enough to Kill You.
In a recent episode of Mythbusters, Brian and Jon (the new MythBusters) wanted to see what happens when you shoot an arrow straight up into the air. It will obviously come back down—but would it still be moving fast enough to kill you? If you are a MythBuster, the best option is to actually shoot an arrow straight up into the air and measure its velocity on impact. But for normal people, it might be better to just calculate the final arrow speed. Also one note—I'm currently a science consultant for the show.
OK, let's start with a basic physics problem. How fast would an object be moving if you dropped it from a height of 300 feet (91.4 meters)? When an object is in free fall, there is only the gravitational force acting on it. Since both the acceleration and the gravitational force depend on the mass, all objects fall with the same acceleration (regardless of mass) with a rate of g = 9.8 m/s2. Of course, this is only true if you ignore air resistance—but we will get to that soon.
If the object starts at a vertical postion y = h and ends at y = 0 meters with a starting speed of 0 m/s, then the final velocity can be found with the following equation. If you want more details on kinematics, here is a resource.
Using the starting height of 91.4 meters, I get a final velocity of 42.3 meters per second (which would be 94.6 mph). So right there you can see that this is a fairly nice estimation of the motion of the arrow. But that analysis leaves off one important aspect—air resistance. As an object moves through the air, there is a force pushing back in the opposite direction of the object's velocity. The magnitude of this air resistance force increases with velocity and also depends on the air and the size of the object. This air resistance force can be written as the following equation: