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to the horizon; as in the projection of balls and shells from mortars, or other pieces of ordnance.
189. Let a body be projected from A, in any direction, not vertical as AD; and let AC, AD, be the spaces that the body would describe in the times t and t', from the uniform velocity of projection, and CE and DB the spaces through which it would descend by the action of gravity in the same times; then it is obvious, from the composition of the two motions that the body will be found at the ends of those times in the points E and B.
Now, by the laws of uniform motions (art. 26) AC=tv, and AD=t'v (v being the velocity of projection), therefore,
AC: AD::t: ť ;
and, by the laws of falling bodies (41), st, and therefore
CE: DB:: t2: t'2
whence CE: DB:: AC: AD2,
which is a known property of the parabola; and, as the same has place for every point in the path of the projectile, it follows, that the curve described by the projectile is a parabola.
190. In order now to investigate the laws of the bodies' motion,
let A be the point of projection AB, or AB', the plane on which the body is projected, passing through A, and which also denotes the range. Let AC be drawn parallel, and BCD perpendicular to the horizon;
let the angle of elevation CADA, and the angle of inclination of the plane CAB B, the velocity of projection =v, the time of flight = 1, and the range AB = r, also let 161⁄2 feet = g.
Then we know from the laws of uniform motion, that the body at the end of the time t, if gravity did not act, would be found in the point D; while by the laws of falling bodies, it would in the same time pass through the perpendicular DB; consequently
Whence we have
AD tv; and DB=g
ABD: sin. BAD :: AD: DB
And equating the value t in equations 1 and 2, we obtain
From these three equations, all the relations between the time, velocity, range, and angle of elevation, are readily determined. For example :
If the time and elevation be given to find the velocity and range; in this case,
gt cos. B
Equation 1 gives v
Equation 2 gives r=
If the range and elevation be given to find the time and velocity we have from
g cos. A
rg cos. B sin. (A+B) cos.
If the velocity and elevation be given, to find the time and range, we obtain from
If any two of the above quantities are given to find the angle elevation we must substitute, instead of sin. (A+B), its value: viz.
sin. (A+B)=sin. A cos. B+ sin. B cos. A
whence sin. A, or cos. A may be obtained.
In all these cases, we shall find sin. (A+B) cos. A equal to a known quantity, which let be denoted by C, then
cos. A sin. A cos. B+sin. B cos. 2A=C
Let sin. A=r, then cos A= √(1−x2); and we have
A quadratic equation from which two values of r will always be determined; and whence we learn that there are always two angles of elevation, which equally answer the conditions of the Problem.
191. To find the greatest height of the projectile above the point of projection, we must observe that the body will continue to ascend till the velocity of des ent from gravity, is equal to the uniform velocity of ascent from projection; that is, calling the time at which the height is greatest t, we sl all have (41) 2gt=the velocity of descent from gravity; and v sin. A will denote the uniform velocity of ascent estimated in a direction perpendicular to he horizon, whence we have
2gtv sin. A, or
v sin. A
but the descent in the time is gt', in whic substituting the above value of ', we have
and the ascent in the same time from projection, is t'u sin. A, or substiuting for as above, we have
and consequently, he difference of these will be the greatest height of the projectile above the point A ; calling therefore the height h, we have
whence, the angle of elevation and the velocity being given, the greatest height will be immediately determined; and if r or t be given, the value of v2 may first be determined from the proper equation, and then the value of h from equation (IV.)
192. All the preceding equations are rendered much if we suppose the plane AB to become horizontal angle Bo, and consequently sin. Bo, and cos. B=1. reduction,
more simple For then the After this
which formulas involve all the conditions of a projectile while the plane is horizontal, and passes through the point of projection.
When the plane is not horizontal, it is obvious that in the formula in which sin (A+B) occurs, B must be accounted positive when the plane descends, and negative when A ascends.
193. If the velocity of the projectile in any point of the curve be required, or the velocity after any time, it is obvious that this is compounded of the constant horizontal velocity of projection v cos. A, and the difference of the vertical velocities of ascent from projection =v sin. A, and that of descent from gravity =2g, that is, calling the required velocity V, we shall have
V=/v2 cos. A+ (v sin. A−2g't) (Equation V.)
Examples on a Horizontal Plane.
1. Let the angle in which a body is projected by 45°, and the time of its flight 12"; what is the horizontal range?
2. In what time will a shell range 3250 feet, in an elevation of 32".
3. If the elevation of the piece be 30° and the horizontal range 2000, what is the greatest height to which the ball will ascend?
How far will a shot range on a plane which descends 8° 15′, the projectile velocity being 440 feet per second, and the elevation of the piece 32° 30' ?
whence the range is 6784 feet nearly.
The learner will observe, that in the preceding reasonings, no notice has been taken of the effects of the resistance of the air on the motion of projec tiles. Now, as this is very considerable, especially when they are discharged with great velocity, the theory requires to be modelled and corrected by experimental investigations, before it can be applied in practice. There are indeed some cases, such as in the throwing of shells, when the velocity does not exceed 400 feet per second, in which the results by the theory do not differ much from the truth. But when the velocity is great, the resistance of the air occasions a diminution of motion so prodigions, as to render the theory, with ont the aid of data derived from experiment, of very little use. Thus, a musket ball, discharged with the ordinary allotment of powder, issues from the piece with the velocity of 1670 feet per second. At the elevation of 45°, it should therefore range 16 miles, whereas it does not range above half a mile. Thus, also, a 24 lb. ball, discharged with 16 lbs of powder, which should range about 16 miles, does not range 3 miles.
Again, the path of a projectile, when the velocity is great, is not parabolical: but is much less incurvated in the ascending than in the descending bract