Measuring the Velocity of Falling Bodies
المؤلف:
GEORGE A. HOADLEY
المصدر:
ESSENTIALS OF PHYSICS
الجزء والصفحة:
p-40
2025-10-23
34
(a) The Direct Method consists of dropping a small ball of some heavy material from the top of a tower - like a shot tower - and determining by actual measurement where it strikes a support at the end of the first second, second second, etc. One of the difficulties connected with this method is the height of tower required, since for a fall of 3 sec. the tower would need to be about 145 ft. high.
(b) Galileo's Method. In all other methods the velocity of the falling body is reduced in some way. Galileo accomplished this by letting a ball roll down an inclined plane. If the length of the plane is made great in comparison with the height, the ball will roll down the plane far more slowly than it would fall in a vertical direction.
If the experiment is carefully made, the results will be such as. are shown in Table A, since the resistance of the air is slight. Let A 1 (Fig. 1), the space passed over in the first second, be called d. Then it will be found that 4 2, the space passed over in 2 seconds, is 4 times as great, or 4 d; that A3, the space passed over in 3 seconds, is 9 times as great, or 9d, etc., no matter what the proportional height of the plane is. These results are shown in the fourth column of Table А, from which are found the spaces passed over in the different seconds as shown in the third column. Since the force is a constant one (it is a certain fraction of the weight of the ball), the acceleration is constant, and is twice the distance passed over in the first second, or 2 d, per second per second. Notice that this is also the difference between any two successive values in column 3. The acceleration 2 d in turn gives the values in the second column.
By increasing the proportional height of the plane, the velocity of the ball is increased, until, when the plane becomes vertical, the ball is no longer a rolling but a falling body and the acceleration equals g, the acceleration due to gravity. Replacing a in Formulas 3, 6, and 4 by g, we have the formulas for falling bodies:
The value of g varies at different places on the earth, from about 978 cm. at the equator to about 983 cm. at the poles. At New York its value is 980.2 cm., or 32.16 feet. By making this substitution, these formulas may be written
These formulas for freely falling bodies are very important, and should be familiar to every student.
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