Note that since we are ignoring friction, it does not matter how long the hill is, only that the height can increase by 40 meters. All of these calculations are examples of the kind of bookkeeping one can do using the first law of thermodynamics.
Appendix: Molecular Pressure and Speeds
It was stated in Section B2 that the average speed of air molecules at room temperature is 1000 miles per hour. Using the concepts developed in Section B2 and in Chapter 3, we can verify that statement. Let us consider a box full of air at normal temperature and pressure.
A molecule of mass m moving with velocity v, has momentum mv and kinetic energy mv2/2. On the average, one-third of the kinetic energy may be associated with motion in any one direction (because space is three dimensional). Because kinetic energy is directly proportional to v2, it must be that the average v in any one direction is 1/V3 of the total v. Hence the momentum of a molecule, which is directly proportional to v, in that direction is mv/V3. When the molecule hits the wall, it will bounce off and move away from the wall with the same speed and momentum that it had before striking the wall, except that now the momentum is in the opposite direction. Hence the change in momentum, which is a vector quantity, is
Now this molecule will move rapidly over to the other side of the box, bounce off that wall, and return again to the first wall. The time interval between collisions with the first wall is At = 2L/(v/V3) where L = length of one side of the box and v/V3 is the average velocity in one direction. The number of collisions by this one molecule per second is 1/At. (If it takes Va sec between collisions, there are 4 collisions per second.)
Now according to Newton's second law (see Chapter 3), the applied force equals the change in momentum per unit time, that is,
In this case, the applied force is the force this one molecule exerts on the wall. This force is then
If there are N molecules in the gas, then the total force is
Now pressure is force per unit area, and the area of the wall is L2. Hence
But L3 is the volume of the box, and Nm is the total mass of the gas, M. Hence we have,
From this expression we can estimate the speed of a molecule of air (air is mostly nitrogen and oxygen). We have
For air at room temperature, P = 14.7 lb/in2 = 105 N/m2, and M/V = density — 1.2 gm/liter = 1.2 kg/m3. Thus,
So we see that air molecules are moving about 1000 mph!
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