920 



SCIENCE 



[N.S. VoT.. XXX. Xo.782 



The velocity at the end of each sec- 

 ond is y = 32.2 ft. per second X 1 2 3 4 .5 



The increase of velocity per second 



is 32.2 ft. per second X 11111 



Velocity, V, is here defined as the rate of mo- 

 tion. It is the space traversed divided by the 

 time, 8 -^T,\i the velocity is uniform. If it is 

 not uniform, but increases at a constant rate, 

 as in falling bodies, then the average velocity 

 during any time, T, is S -h T. If the velocity 

 is at the beginning of any time T and Y 

 at the end of the time, then V = 'iS-^T. 

 The relation of time, space and velocity when 

 the velocity is uniformly increasing may be 

 illustrated by a right-angled triangle in which 

 the bases are T and V and the area 8. 



S = iVT, V = 28-i-T (1) 



Expressing algebraically the results given in 

 the above table we find 

 Total fall, S = 16.1 xr-, or if 



(7 = 32.2, S = i5rr (2) 

 Velocity at the end of the time T, 



V = 32.ZXT — gT (3) 

 Velocity at the end of the fall 8, 



V = V2">r32.^2" X 8 ^^2p( { 4 ) 

 The last equation is commonly written 

 V=\/2gE, in which fl' = height of fall. 



Acceleration.— A, = 7 -=- T = rate of in- 

 crease of velocity, := 32.2 ft. per second in each 

 second in the case of falling bodies at London. 

 This quantity, 32.2 ft. per second per second, 

 is commonly represented by g, and it is called 

 the acceleration due to the earth's gravitation, 

 or more briefly, the acceleration due to gravity. 



Force causing Acceleration. — In the ease of 

 a falling body, a force which equals the weight 

 W acts on the weight and causes it to move 

 with an acceleration A = 32.2 ft. per sec. per 

 sec. Suppose the weight is not a falling body, 

 but a weight supported on frictionless rollers 

 on a level plane, or a body floating in still 

 water, and a constant force is applied to it 

 horizontally, say by a cord the tension in 

 which is kept constant and measured by a 

 spring balance. What then will be the accel- 

 eration? The force is the cause of accelera- 

 tion, and the acceleration is the effect of force. 

 In general an effect is proportional to its 

 cause. If the force applied to the weight is 



one tenth of the force of gravity, then the 

 acceleration is one tenth that which would be 

 caused by gravity, or 0.1 g or 3.22 ft. per sec. 

 per sec. 



Let T'F = weight, g<^the acceleration that 

 would be given by a force equal to W, 4 = the 

 acceleration that would be given by any other 

 force F, then 



a: g-.: f:w, 



or the acceleration produced by any force F 

 is to the acceleration due to gravity as the 

 force F is to the weight of the body, or 



A = gF/W. (5) 



From this equation we see that the accelera- 

 tion is proportional to the force, and inversely 

 proportional to the weight. Illustrate this by 

 Atwood's machine and by experiments on 

 weights supported on rollers or floating in 

 water. Writing A instead of g in the equa- 

 tions (2), (3) and (4), they become general 

 and apply to all cases of uniformly accelerated 

 motion, as well as to falling bodies. 



Total space traversed in time T, S^^iAT' (6) 



Velocity at the end of the time T,V^AT (7) 

 Velocity at the end of the space S, 



V = V2A8 (8) 



Example (1), A boat in a canal weighs 20,000 

 pounds. If a boy pulls it with a string with 

 a constant pull of 10 pounds, what velocity will 

 the boat have acquired at the end of a minute, 

 friction being neglected? Ans. 0.966 foot per 

 second. (2) If an air gun has a bore 5 feet 

 long and 1 square inch area, a bullet weighing 

 one pound sliding frictionless in the bore, and 

 propelled by compressed air supplied from a 

 large reservoir at a constant pressure of 2,000 

 pounds per square inch behind the bullet, 

 what will be the velocity of the bullet as it 

 leaves the gun? Ans. 802.5 feet per second. 

 In the equation (5) A=gF/W, F and W 

 are known and A is to be found, but if the 

 acceleration is known, and also either one of 

 the other two quantities, F or W, the third 

 quantity may be found. Thus, 



WA = gF, whence W = gF/A 



(9) 



and 



F^W/gXA. (10) 



That is if a force produces a uniform accel- 



