6 4 KINEMATICS. [126. 



Careful experiments on the resistance offered by the air to 

 the motion of projectiles have shown that this resistance in- 

 creases with the quantity of air displaced ; that is, with the 

 density of the air, the cross-section of the projectile, and the 

 velocity. The retardation due to the resistance of the air can 

 therefore be expressed in the form 



j=>cpf(v), 



where p is the density of the air, while K is a coefficient depend- 

 ing upon the shape, mass, and physical condition of the surface 

 of the projectile. Its value may be regarded as inversely pro- 

 portional to the mass and directly proportional to the cross- 

 section of the body at right angles to the direction of motion. 

 The velocity function/^) may be taken =cv* for velocities 

 not exceeding 250 metres per second ; for greater velocities, up 

 to about 420 metres per second, it is proportional to a higher 

 power of v, or must be represented by a more complicated 

 expression, such as aiP -\-bv-\-c\ for velocities above 420 metres 

 it seems to be again of the form c'v 2 .* 



126. Assuming the resistance of the air to be proportional to 

 the square of the velocity, the motion of a body falling through 

 air of uniform density is determined by the equation 



To simplify the resulting formulae, it will be convenient to 



U? 



put K = , so that the equation of motion is 



, } 



dft g 



Writing -^ for |, the variables v and t can be separated : 

 at dp 



=dt; 



* For further particulars the reader is referred to special works on ballistics. 



