which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion.
| Fig. 121. |
Consider the possible changes of value of the fraction
(1 − ζu2 / 2gih) / (1 − u2 / gh).
As h tends towards the limit 0, and consequently u is large, the numerator tends to the limit −∞. On the other hand if h = ∞, in which case u is small, the numerator becomes equal to 1. For a value H of h given by the equation
1 − ζu2 / 2giH = 0,
H = ζu2 / 2gi,
we fall upon the case of uniform motion. The results just stated may be tabulated thus:—
For h = 0, H, > H, ∞,
the numerator has the value −∞, 0, > 0, 1.
Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit −∞. As h becomes very large and u consequently very small, the denominator tends to the limit 1. For h = u2/g, or u = √ (gh), the denominator becomes zero. Hence, tabulating these results as before:—