# Real vs. Ideal Stones

*Most recent answer: 10/22/2007*

Q:

If you are at the top of a cliff, and want to measure the height of the cliff practically, is it possible to calculate the height by dropping a stone off, measuring the time until it drops, and plugging the time into the equation S=(1/2)at^2? Is that the correct equation to use (if a=9.8m/s^2), and are there simply too many variables involved to make an accurate measurement(wind, air resistence, etc.)?

- Anonymous

- Anonymous

A:

Well, there several things that determine whether your answer will be accurate or not.

I’ll start with air resistance. There are 2 major things that will affect air resistance. The first is the size of the stone. The cross sectional area of the stone will determine how much air resistance it will experience. However, the size of the stone also determines how heavy it is. Air resistance (discussed below) will be a problem for your measurement if it gets anywhere near as big as the weight of the rock. However, since the weight of the rock will be proportional to its radius cubed, and the area of the rock to its radius squared, making the rock bigger is good since it will make air resistance relatively smaller than gravity.

The second part is the velocity of the stone. The cool thing about air resistance is that it depends on how fast you are moving. That makes sense if you think about it. You have to push through a lot more air in the same time if you’re moving faster. The velocity of the stone depends on how tall the cliff is. If the cliff is very very very tall, then the air resistance will eventually equal the force of gravity and then your acceleration will be 0 instead of 9.81. That is called the "terminal velocity" because it won’t get any faster.

Then you mention the wind. Depending on how the wind is moving and the shape of the cliff, the wind could actually flow up the cliff. That could help slow the rock down a little bit if the wind is strong enough.

So if your cliff isn’t too high, and there isn’t too much wind, and you use a big rock, then you have a good way to tell the height of the cliff.

Adam

I’ll start with air resistance. There are 2 major things that will affect air resistance. The first is the size of the stone. The cross sectional area of the stone will determine how much air resistance it will experience. However, the size of the stone also determines how heavy it is. Air resistance (discussed below) will be a problem for your measurement if it gets anywhere near as big as the weight of the rock. However, since the weight of the rock will be proportional to its radius cubed, and the area of the rock to its radius squared, making the rock bigger is good since it will make air resistance relatively smaller than gravity.

The second part is the velocity of the stone. The cool thing about air resistance is that it depends on how fast you are moving. That makes sense if you think about it. You have to push through a lot more air in the same time if you’re moving faster. The velocity of the stone depends on how tall the cliff is. If the cliff is very very very tall, then the air resistance will eventually equal the force of gravity and then your acceleration will be 0 instead of 9.81. That is called the "terminal velocity" because it won’t get any faster.

Then you mention the wind. Depending on how the wind is moving and the shape of the cliff, the wind could actually flow up the cliff. That could help slow the rock down a little bit if the wind is strong enough.

So if your cliff isn’t too high, and there isn’t too much wind, and you use a big rock, then you have a good way to tell the height of the cliff.

Adam

*(published on 10/22/2007)*