132 
MOTION OF A PEOJECTILE 
Suppose the resistance to vary as the n ih - power of the velocity, then 
the retardation due to the resistance will he g /- j ; w being the ter¬ 
minal velocity in the medium. 
To express % in terms of p 3 resolve horizontally; therefore 
du /v\ n 
cos +‘ 
i . dt u 
and since - r = -— > 
dp g 
therefore t- = (—i » cos d/ 
dp \wj 
u n+1 
or «**+1 dp 1 + p ’ ’ 
Integrating, supposing the velocity is infinite when the trajectory is 
carried back to the point where the tangent of the inclination to the 
horizon is a , 
©•-* . 
_ i 
or u = wP », .....(4) 
a m -1 
where P=n f* (1+^ 2 ) 2 dp ...(5) 
-9p 
i _ i 
and therefore » = w (1 + ]?-f P % . 
Therefore ^ = — - P *, 
— = — — p~ ^ 
dp g 
d JL- - ^vP'n. 
dp~ g* ' 
and carrying the origin back to the point at which the velocity is 
infinite, 
t=- f a P~" dp, ....(6) 
9 J v 
x=-f a p'”dp, ...( 7 ) 
9 Jp 
2 
0/12 /"*& — — 
y = — / pP n dp .(8) 
9 Jp 
