VELOCITY OP PROJECTILES.] MECHANICAL PHILOSOPHY. DYNAMICS. 



739 



This is what is called the equation to a parabola, A B 



Fig. 143. 



parallel to C P being a tangent to the curve at the point 

 A, and A C being parallel to the axis. The equation is 

 usually written thus : j/ a =az, where y is the ordinate 

 C P of any P, and x the abscissa A C, of that point, 

 while a is any constant multiplier of the abscissa. The 

 equation when A is at the vertex of the curve, is inves- 

 tigated at page 610 of the PRACTICAL GEOMETRY. It 

 follows, therefore, that the projectile in its flight always 

 traces out a parabola : the constant multiplier in its 



equation is It is known, from the theory of the 



parabola, that this constant multiplier is also 4 S A, 

 S being the focus of the curve : it is further known, and 

 is proved at the page just referred to, that the distance 

 A S of any point A in the curve from the focus, is always 

 equal to the distance of the same point from the directrix. 

 Let this distance be called h, then 



4SA = =*&.-. v>=2gh .... (2) 



y 



But this value expresses the square of the velocity 

 which a body would require, from the force of gravity 

 g, by falling from rest, from the height h ; consequently, 

 the velocity of projection is equal to that which the 

 body would acquire by falling from the directrix of the 

 parabola, which it traces, down to the point of projec- 

 tion. 



VELOCITY AT ANY POINT OF THE PATH. It 

 has just been shown that A being the point of projec- 

 tion, and c the velocity of projection, that velocity will 

 be = V(2<7 ' SA). But any point P of the path may be 

 regarded as the point of projection, and the correspond- 

 ing velocity as the velocity of projection ; so that the 

 expression (2) applies equally to any point in the curve 

 traced, h being the distance of that point below the 

 directrix, and v the velocity at the same point. Hence, 

 the velocity of the projectile at any point of its course is 

 the same as the velocity it would acquire by falling ver- 

 tically from the directrix down to that point. And it 

 further follows, that at equal heights, in ascending and 

 descending, the projectile will have equal velocity. 



GREATEST HEIGHT. In order to ascertain the 

 greatest height to which the projectile will ascend, we 

 must of course know the velocity of projection, and the 

 direction of that velocity. It will also be convenient to 

 replace this velocity by two component velocities equi- 

 valent to it in effect : the one a horizontal velocity, and 

 the other a vertical velocity (see page 738)- If a be the 

 angle of projection, that is, the angle of elevation of the 

 initial direction above the horizontal line, then the velo- 

 city v of projection, will be equivalent to the horizontal 



velocity v. cos. a, combined with the vertical velocity 

 v. sin. a, 



Now, it is plain that gravity, which acts only in a 

 vertical direction, cannot in any way disturb the hori- 

 zontal velocity v. cos. a, so that this velocity is the same 

 at every point of the path. But the vertical velocity v, 

 sin. a, having the whole influence of gravity to check 

 and oppose it, will be utterly destroyed and reduced to 

 0, when gravity has acted sufficiently long to impress on 

 the body (supposing it left free to obey its solicitations) 

 a downward velocity, equal to the upward velocity . 

 sin. a. The spaces through which a body must fall to 

 acquire this velocity, or the height to which it must 

 ascend to lose this velocity, is 



, (velocity ) , r 2 sin. 8 a 



-$- = J = h sin. 2 a (by equa. 2) 



y y 



Hence, the greatest height to which the projectile dis- 

 charged in a given direction can ascend, will be found 

 by multiplying the height (h) through which a body must 

 fall, to acquire the velocity (v) of projection, by the 

 square of the sine of the angle (a) of elevation. 



VVith the same projectile, velocity , the highest point 

 to which the projectile can ascend under different eleva- 

 tions, will of course be that due to the elevation of 90 ; 

 and we see accordingly, that the multiplier sin. 2 a is 

 greater for this value of a, than for any other. 



TIME OF FLIGHT. The time o.cupied in the flight 

 of the projectile, that is, the time from discharging it 

 till it falls to the horizontal plane passing through the 

 point of departure, will of course be double the time 

 in which the vertical velocity is destroyed, as the body 

 must fall to the horizontal plane with the same vertical 

 velocity with which it left it. The time in which, by 

 gravity, the velocity v sin. a would be generated, ia 



velocity v. sin. a 

 9 9 



.'. 2t = 2- ', the time of flight. 



The time of flight, therefore, as might be expected, is 

 the greater (with the same velocity of projection), the 

 nearer the direction of projection approaches to the ver- 

 tical ; since sin. a increase if a increase and does not ex- 

 ceed 90. 



RANGE OF THE PROJECTILE. The range is the 

 distance on the horizontal plane, through the point of 

 departure, at which the projectile falls. It has just 



been seen that the time of flight is 2- 



, and since 



the horizontal velocity v. cos. a is uniform all this time, 

 we have only to find the space due to the velocity v. 



cos. a in the time 2 -- , that is, we have 



Range =r cos. a X 2- 



in. a D 2 X 2 sin. ocos. a 2 . 

 = = sin. 2a 



9 99 



Hence, with the same velocity v of projection, the 

 range increases as sin. 2a increases, and this it does as a 

 increases from a=0 up to a=45 : therefore, with the 

 same velocity of projection, the range at an elevation of 

 45 will be greater than at any other elevation whatever. 

 And since the sine of an angle is the same as the sine of 

 its supplement, that is, since sin. 2a is the same as sin. 

 '180 2o), it follows that the elevation 90 a gives the 

 same range as the elevation a : hence the ranges of pro- 

 jectiles, at any elevations above 45", are the same as the 

 ranges at elevations as much below 45. 



The length of the maximum range, that is, when a= 



45, is shown above to be ; v being the initial velocity, 



and this, as proved above (equation 2), is the same 

 as 2/i : so that the length of the maximum range is equal 

 to twice the height due to the velocity of projection. 



If the range of a shot with a known elevation of the 

 piece be ascertained, it will be easy to determine what 



