Acceleration
المؤلف:
GEORGE A. HOADLEY
المصدر:
ESSENTIALS OF PHYSICS
الجزء والصفحة:
p-37
2025-10-21
38
When the velocity of a body is uniformly accelerated, the rate of change in its velocity or the amount its velocity changes per second, is called its acceleration. If the body, starting from a condition of rest, has a velocity of 2 ft. per second at the end of one second, 4 ft. per second at the end of the next second, 6 at the end of the third, and so on, the gain in velocity, per second, is 2 ft. per second; that is, the acceleration is 2 ft. per second per second. In uniformly accelerated motion the average velocity for any period is half of the sum of the velocities at the beginning and end of that period. Hence in this case, as the velocity is O at the beginning, the average velocity for the first second would be 1 ft. per second, and the average velocity for the first three seconds would be. 3 ft. per second.
Suppose a body to move in a certain direction from a condition of rest with a constant acceleration of a units per second per second. Its velocity per second at the end of 1 sec. will be a; at the end of 2 sec., 2 a; at the end of 3 sec., 3 a; and so on. At the end of t seconds its velocity per second will be t × a or at; that is,
Since the velocity per second increases uniformly from 0 at the beginning to at the end of t seconds, the average velocity per second for t seconds will be 
at, and the entire space passed over equals average velocity × time. This may be represented by the equation
Since the velocity per second for the first second increases uniformly from 0 at the beginning to a at the end, the space passed over in that second will be a. We can get the same result by making the time 1 sec. in Equation 4, for the equation then becomes
a We see from this that whenever a body starts from a condition of rest, and moves with a constant acceleration, the acceleration per second per second is twice the space passed over in the first second.
twice the space passed over in the first second. The space passed over during any second (the last of t seconds) may be found by subtracting from the distance passed over in t seconds the distance passed over in a time one second less. The space passed over in t seconds be-ing by Formula 4,
the space passed over in (t - 1) seconds will be
Hence the space passed over in the last second will be
It is sometimes convenient to express velocity and acceleration in symbols. Thus, a velocity of twenty centimeters per second may be expressed as 20cm/sec, and the speed of a bicycle rider going at the sec. ft. rate of a mile in four minutes is a speed of 22ft/sec that is, twenty-two sec. feet per second. In the same way an acceleration of fifteen centime. ters per second per second may be written 15cm/sec2 and an accelera-. tion of twelve feet per second per second may be written 12ft/sec2.
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