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The path of a projectile
Oct., I89L]
])at/~ o f
a Projectile.
267
its action. W e are, therefore, e n t i t l e d to respect q u i t e h i g h l y this useful combination, whose w o r t h can never be accur a t e l y estimated, since its chief service is to p r e v e n t certain losses and accidents of variable character, and whose e x t e n t is the only m e a s u r e of the loop's value. THE P A T H oF A P R O J E C T I L E . By F. GILMAN, Lowell, Mass.
T h e object of the present article is to show how well t h e e q u a t i o n of a projectiles path, developed on the h y p o t h e s i s t h a t the resistance varies directly as the velocity, c o n f o r m s to the e x P e r i m e n t a l results of the Chief of Ordnance, U.S.A. I shall also s u b m i t the e q u a t i o n to the same test w h e n developed on the h y p o t h e s i s t h a t the resistance varies as the square of the velocity, a l t h o u g h in this case the comparison will necessarily be restricted to e x p e r i m e n t s in which the angle of elevation was very small, As the equation of t h e t r a j e c t o r y d e d u c e d on the a s s u m p t i o n of a resista n t e proportional to t h e velocity, is not g e n e r a l l y given in treatises on m~chanies, we will briefly show how it is derived. L e t the resistance be expressed by c v, in w h i c h v i s the velocity, and c a c o n s t a n t t h a t expresses the resistance for a u n i t of velocity. T h e origin of co6rdinates will be taken at the point from w h i c h the projectile is t h r o w n , x b e i n g the horizontal a n d y t h e vertical axis. L e t t denote the time, a n d g the acceleration of g r a v i t y . Then, according to the f u n d a m e n t a l e q u a t i o n s of m e c h a n i c s : d~x it
dx =
t2Y --
dg
-- c df
g
c -dt-
Let B
b--
and s u b s t i t u t e this in
(I).
ix dt
(1)
(2)
268
Gilman
[J. F. I.,
."
It becomes db dt
-
cb
-
dt --
db cb
(3)
a n d integrating, t--
: logb+C
C
in which C is the constant of integration. I n order to d e t e r m i n e its value, let v denote the velocity of the projectile, and a the initial value angle which its t r a j e c t o r y m a k e s with the axis of x. since the t i m e is e s t i m a t e d from the b e g i n n i n g motion. I o =
--
C
Zos~ ( V cos a) +
initial of the Then, of t h e
C
C = I Zog(Vcos ~) C
whence,
t = : ICo g
( I7 cos a~
-7~/
(4)
Multiplying equation (3) by b, we have, bdt .~ dx =
--
db --
£
a n d i n t e g r a t i n g with due reference to the value o f the c o n stant, V cos a - -
b
£
whence, ~
~
COS
~ --
CX
and s u b s t i t u t i n g this value of b in the above expression for 1, we obtain, t = - I log ( U .cos,z ... ) (5) c
U cos a - -
ca;
T h i s equation gives the time required for the projectile to reach a n y point i n . i t s p a t h whose horizontal distance
Oct., I89I. ]
269
Pat]~ of a Projectzle.
from the origin is equal to x. tion (2), substitute b for
In order to integrate equa-
dt
and proceed exactly in the same manner as for equation (i). W e find, t_~_ gt log ( g +g c3:V ~sin a )
(6),
Equations (4) and (6) give the time required for the projectile to reach any point in its path. Equating them and restoring the values of b :
---dZ-
=
zog
\
~
ei
)
Passing from logarithms to quantities, taking the reciprocals and multiplying through by dr, dac _ g d t + cdy U cos a g 4- c V sin a
Integrating, substituting for t its value from (5), and solving with respect to y, Y--g4-
c V s i n aac g log ( Vcos a c V cos a - - -~ V cos-a- c x )
(7)
which is the equation of the trajectory. An analysis of this equation shows that a perpendicular to the axis x at a distance from .the origin equal to 17 cos tz ¢
will be an asymptote to the curve. Differentiating the equation, and finding the value of ~from -* d.Y __
0
dac
we obtain U cos a {
c
c U sin a
\
k j g c-~ y,,~ )
173 sin a cos cx
~ + 5 7 s~n
270
G i l m a n ."
I
! J. F. I.,
which gives the horizontal distance from (he origin at which the projectile attains its g r e a t e s t elevation. T h i s value of x s u b s t i t u t e d in e q u a t i o n (5) gives t' = I log ( C V sin a + g ) c g
w h i c h gives the t i m e in w h i c h the projectile will reach its g r e a t e s t h e i g h t . T h e expresgion for this h e i g h t is Y = I_ ( V s i n
a--gt
I)
C
F r o m e q u a t i o n (7) we deduce the following formu!a for the range R~ g Ucosat g + c V sin a T h e value of c depends on the initial velocity, the n a t u r e of t h e projectile, and the angle of elevation at which it is thrown. For each class of projectiles, and for each initial velocity, t a k e n at intervals of from 50 to IOO feet per second, t h e r e will be a certain curve whose abscissas will r e p r e s e n t the angles of elevation, and its ordinates t h e c o r r e s p o n d i n g values of c. Fz~. z shows such a curve, w h i c h gives the values of c in the e x p e r i m e n t s m a d e by the Chief of O r d n a n c e on the firing of two guns, and recorded in his report for i885, pp. 66, 495. One of the g u n s was a fifteen-inch R o d m a n s m o o t h bore. T h e projectile was r o u n d iron solid shot, and its w e i g h t 455 pounds. T h e initial velocity was 1,678 f. s. T h e other g u n was a 3"2 inch breech-loading rifle. T h e projectile was a solid b a n d e x p e r i m e n t a l shell, and its w e i g h t t h i r t e e n pounds. T h e initial velocity was 1,6o2 f. s. T h e values of c, p l o t t e d from these two sets of experiments, g a v e two curves, w h i c h ran so n e a r l y together, t h a t it was t h o u g h t best to combine t h e m into one m e a n curve, prov i d i n g t h a t the c o m p u t e d results w o u l d agree w i t h t h e experiments, as well as the e x p e r i m e n t s agree w i t h themselves.
Oct.,
I891.]
2DOf/'/Z o f
a Projectile.
271
T h e following table gives a synopsis of the results : ~5-INCH RODMAN
A n g l e of
V a l u e of c t a k e n frc, m
Elevation.
Diagrc m,
Fig. z,
GUN.
RAl~c-~
T I M g oF FLIGHT
" i! I By E x p e r i m e n t . l B y F o r m u l a . B y Experiment.: B y F o r m u l a .
fetl.
feet.
seconds.
• 2o
0'0673
5,300
5,23 °
4'0
5°
O'IZl2
8,857
8,8*0
8'i
7"9
Io 0
0'0890
73,696
13,636
I3" 9
r4' 9
i5 °
o'o748
16,65 I
I7,3z9
i9"i
2I' 5
~oo
o'o66o
x9,285
20,043
23"8
~7'7
23°
o"o622
21,243
2~,268
26"4
3*'2
3'2-1NCH BREECH-LOADING
Angle of Elevation.
2o
V a l u e of c t a k e n from Diagram,
F i g . I.
o'o673
I I [
seconds. 3"5
RIFLE.
RANOE
TIME
OF
FLIGIIT
]By E x p e r i m e n t .
fly F o r m u l a .
By E x p e r i m e n t .
By F o r m u l a .
feet.
feel. 4,797
seconds.
second. 3"3
6'7
4,755
4'o
6"2
4°
o'*I48
7,o93
7,096
6o
o'Io65
9,xo9
9,215
9'o
9'o
8°
0'0974
zo,9o7
ii,o66
Ii' 5
II" 7
io o
0"0890
x2.45o
I2,77o
13' 7
14"3
12°
o'e818
I4,472
I4,354
15' 7
x6' 9
o'o77 o
z5,8io
I5,658
x7' 9
19"4
16°
0"0725
I7~O7O
~6,89o
2o
ex" 9
18°
0"0690
x8,29~
I7,944
22
~4".3
2oo
0'066o
I9,437
I8,87o
24
26"6
14°
.
T h e fourth and fifth of the R o d m a n experiments s h o w the greatest discrepancies, a m o u n t i n g to about four per cent. of the range. It will be found, however, on e x a m i n i n g the separate experiments of which these are the mean, that they show differences a m o n g t h e m s e l v e s of from seven to eleven per cent. T h e initial velocity was m e a s u r e d only in the first few experiments, and it was a s s u m e d to be the same in all the others, as the w e i g h t and kind of powder was the same. But it is probable that this a s s u m p t i o n is
272
Gilman :
[J. F. I.,
n o t s t r i c t l y correct and it w oul d h a v e b e e n desi rabl e to h a v e h a d a m e a s u r e m e n t of t h e initial v e l o c i t y at each e x p e r i m e n t , as it seems likely t h a t t h e differences in t h e r a n g e s c o r r e s p o n d i n g to t h e s a m e a ngl e of e l e v a t i o n are p r i n c i p a l l y due to differences in t h e initial velocities. T h e c c u r v e g i v e n in Y@. z of a c c o m p a n y i n g plate, is pecul i ar in t h a t it shows s uch a g r e a t v a r i a t i o n in t he v a l u e of c f o r angles of e l e v a t i o n b e t w e e n 2 ° and 4 ° . B e y o n d 5°, as the a ngl e of e l e v a t i o n increases, t h e v a l u e of c g r a d u a l l y d i m i ni s he s , a n d t h e c u r v e is v e r y r e g u l a r . T h e c u r v e was c o n s t r u c t e d so as to gi ve as c o r r e c t v a l u e s as possible for t h e ranges, and w i t h o u t r e g a r d to t he times, w h i c h wer e left to c om e out as t h e y would. It will b e n o t i c e d t h a t th e c o m p u t e d t i m e s are s m a l l e r t h a n t h e o b s e r v e d for angles of e l e v a t i o n of f r o m 2 ° to 5 ° , t h a t f r o m 5 ° to 6 ° t h e o b s e r v e d and c o m p u t e d t i m e s are p r a c t i c a l l y t h e same, while b e y o n d 6 ° t h e excess of t he c o m p u t e d o v e r t h e o b s e r v e d v al ues g r a d u a l l y i ncr e a s es w i t h t he angl e of elevation. If it were r e q u i r e d to c o m p u t e t he t i m e w i t h g r e a t e r exactness, let t' d e n o t e t h e c o r r e c t e d value, t the v a l u e o b t a i n e d d ir ectl y f r o m f o r m u l a (5), and d e n o t i n g b y n t h e a n g le of e l e v a t i o n ( e xpr es s ed as seconds), we will h a v e f o r t h e R o d m a n gun, t ' = t--(Tn~) 2 and for t h e 3"2:inch rifle, t' =
t - - ~g.
T h e s e f o r m u l a s will a p p l y w h e n t h e a ngl e of e l e v a t i o n e x ceed s 8 °. F i g . ~ shows t h e c c u r v e for e x p e r i m e n t s m a d e w i t h t h e forty-two m i l l i m e t r e H o t e h k i s s m o u n t a i n gun, and r e c o r d e d in th e O r d n a n c e R e p o r t for I885, p. 88. T h e initial v e l o c i t y was a b o u t 1,3oo f. s., and the w e i g h t of t h e projectile, one p o u n d fifteen ounces. F i g . 3 r e p r e s e n t s t h e p a t h of a p r o j e c t i l e as d e d u c e d fr o m d a t a of one of t he R o d m a n g u n e x p e r i m e n t s a b o v e r e c o r d ed , t h e angl e of e l e v a t i o n b e i n g 23 ° , and t h e val ue of c o'o622. T h e v a l u e s of t h e o r d i n a t e s w e r e c a l c u l a t e d f r o m e q u a t i o n (7) fbr e v e r y 5oo f e e t of h o r i z o n t a l distance, and the r e s u l t s i~lotted, give t h e curve. It is u n s y m m e t r i c a l in form, and flatter on t he a s c e n d i n g t h a n on t he d e s c e n d i n g side, as is known_ by o b s e r v a t i o n to be t he case, in t h e
Oct., 189I.]
Pat,lz of a Projectile.
~73
a c t u a l p a t h of a projectile. T h e horizontal distance from t h e origin at w h i c h the projectile a t t a i n s its g r e a t e s t elevation is, in this example, a b o u t two-thirds of the range. In order to show how well these ordnance e x p e r i m e n t s s u s t a i n the h y p o t h e s i s of resistance proportional to the square of the velocity, we wili take the equations t h a t apply to this case, and which are g e n e r a l l y given in treatises on mechanics. t
--
I
(6 c r -
I)
6 [/r60S 8 62 ~1 6 0 S 2 ~'t" l a * l (~ -~2 g2cr _ _
2 67" - -
I
In these equations t represents the t i m e of flight, a the angle of elevation, z, the initial velocity, r the range, h tile h e i g h t due to the initial velocity, and c the c o n s t a n t of resistance. Since these equations can be applied only to cases in w h i c h the angle of elevation is v e r y small, we will select for comparison the first two of the R o d m a n g u n experiments. W e find t h a t the value of c, which is o b t a i n e d by substit u t i n g for r the correct range, is in the first e x p e r i m e n t o'oooo386, and in the second o'oooo8io. T h e corresponding realness of the times are 3"5 seconds and 7"7 seconds. W e see t h e n t h a t for angles of elevation b e t w e e n 2 ° and 5 o c is even more v a r i a b l e t h a n was f o u n d to be the case on the h y p o t h e s i s of a resistance proportional to the velocity, .'rod as the constancy of the value of c (at least for small angles of elevation), would seem to be a good test of the accuracy of the h y p o t h e s i s on w h i c h it is based, it follows t h a t the a s s u m p t i o n of a resistance proportional to the square of the velocity is no b e t t e r s u s t a i n e d by these'experiments, t h a n was the a s s u m p t i o n of resistance directly as the velocity. T h e values of c for the first two e x p e r i m e n t s w i t h the 3"2 inch breech-loading rifle, are 0"ooo0486 and o'oooo355 , and the c o r r e s p o n d i n g values of the times 3"3 and 6"t. It appears t h a t the resistance of the air can h a r d l y be expressed in t e r m s of a n y s i n g l e power of the velocity, b u t VOL. CXXXII. x8
274
Gillnan.
[J. F. I.,
I t h i n k it can be expressed quite a c c u r a t e l y by the following formula : R =
a + b y -t- c v ~
in which 1~ is the resistance, v the velocity, and a, b and c three constants to be d e t e r m i n e d by experiment. I h a v e applied this f o r m u l a to the e x p e r i m e n t s of H u t t o n on air resistance, and after d e t e r m i n i n g the values of the c o n s t a n t s b y least squares, f o u n d t h a t it satisfied those e x p e r i m e n t s w i t h i n the limits of probable errors of observation. T h e r a n g e of the velocities was from 95o to 2,o5o feet per second. T h e g r e a t e s t difference b e t w e e n the c o m p u t e d and experim e n t a l resistance was a b o u t three per cent., and the m o s t of t h e m were m u c h smaller. But, a l t h o u g h this f o r m u l a p r o b a b l y represents the law of resistance, y e t to a t t e m p t to m a k e it the basis of a m a t h e m a t i c a l d e d u c t i o n for finding the e q u a t i o n of a projectile's p a t h w o u l d be a hopeless problem, from its complexity. It seems more practicable to find the e l e m e n t s of the trajec_ t o r y by the first m e t h o d shown above; especially since the r e s u l t s of t h a t m e t h o d are as accurate as those arrived at on t h e g e n e r a l l y accepted h y p o t h e s i s of resistance propor tional to the square of the velocity.
])at/~ o f
a Projectile.
267
its action. W e are, therefore, e n t i t l e d to respect q u i t e h i g h l y this useful combination, whose w o r t h can never be accur a t e l y estimated, since its chief service is to p r e v e n t certain losses and accidents of variable character, and whose e x t e n t is the only m e a s u r e of the loop's value. THE P A T H oF A P R O J E C T I L E . By F. GILMAN, Lowell, Mass.
T h e object of the present article is to show how well t h e e q u a t i o n of a projectiles path, developed on the h y p o t h e s i s t h a t the resistance varies directly as the velocity, c o n f o r m s to the e x P e r i m e n t a l results of the Chief of Ordnance, U.S.A. I shall also s u b m i t the e q u a t i o n to the same test w h e n developed on the h y p o t h e s i s t h a t the resistance varies as the square of the velocity, a l t h o u g h in this case the comparison will necessarily be restricted to e x p e r i m e n t s in which the angle of elevation was very small, As the equation of t h e t r a j e c t o r y d e d u c e d on the a s s u m p t i o n of a resista n t e proportional to t h e velocity, is not g e n e r a l l y given in treatises on m~chanies, we will briefly show how it is derived. L e t the resistance be expressed by c v, in w h i c h v i s the velocity, and c a c o n s t a n t t h a t expresses the resistance for a u n i t of velocity. T h e origin of co6rdinates will be taken at the point from w h i c h the projectile is t h r o w n , x b e i n g the horizontal a n d y t h e vertical axis. L e t t denote the time, a n d g the acceleration of g r a v i t y . Then, according to the f u n d a m e n t a l e q u a t i o n s of m e c h a n i c s : d~x it
dx =
t2Y --
dg
-- c df
g
c -dt-
Let B
b--
and s u b s t i t u t e this in
(I).
ix dt
(1)
(2)
268
Gilman
[J. F. I.,
."
It becomes db dt
-
cb
-
dt --
db cb
(3)
a n d integrating, t--
: logb+C
C
in which C is the constant of integration. I n order to d e t e r m i n e its value, let v denote the velocity of the projectile, and a the initial value angle which its t r a j e c t o r y m a k e s with the axis of x. since the t i m e is e s t i m a t e d from the b e g i n n i n g motion. I o =
--
C
Zos~ ( V cos a) +
initial of the Then, of t h e
C
C = I Zog(Vcos ~) C
whence,
t = : ICo g
( I7 cos a~
-7~/
(4)
Multiplying equation (3) by b, we have, bdt .~ dx =
--
db --
£
a n d i n t e g r a t i n g with due reference to the value o f the c o n stant, V cos a - -
b
£
whence, ~
~
COS
~ --
CX
and s u b s t i t u t i n g this value of b in the above expression for 1, we obtain, t = - I log ( U .cos,z ... ) (5) c
U cos a - -
ca;
T h i s equation gives the time required for the projectile to reach a n y point i n . i t s p a t h whose horizontal distance
Oct., I89I. ]
269
Pat]~ of a Projectzle.
from the origin is equal to x. tion (2), substitute b for
In order to integrate equa-
dt
and proceed exactly in the same manner as for equation (i). W e find, t_~_ gt log ( g +g c3:V ~sin a )
(6),
Equations (4) and (6) give the time required for the projectile to reach any point in its path. Equating them and restoring the values of b :
---dZ-
=
zog
\
~
ei
)
Passing from logarithms to quantities, taking the reciprocals and multiplying through by dr, dac _ g d t + cdy U cos a g 4- c V sin a
Integrating, substituting for t its value from (5), and solving with respect to y, Y--g4-
c V s i n aac g log ( Vcos a c V cos a - - -~ V cos-a- c x )
(7)
which is the equation of the trajectory. An analysis of this equation shows that a perpendicular to the axis x at a distance from .the origin equal to 17 cos tz ¢
will be an asymptote to the curve. Differentiating the equation, and finding the value of ~from -* d.Y __
0
dac
we obtain U cos a {
c
c U sin a
\
k j g c-~ y,,~ )
173 sin a cos cx
~ + 5 7 s~n
270
G i l m a n ."
I
! J. F. I.,
which gives the horizontal distance from (he origin at which the projectile attains its g r e a t e s t elevation. T h i s value of x s u b s t i t u t e d in e q u a t i o n (5) gives t' = I log ( C V sin a + g ) c g
w h i c h gives the t i m e in w h i c h the projectile will reach its g r e a t e s t h e i g h t . T h e expresgion for this h e i g h t is Y = I_ ( V s i n
a--gt
I)
C
F r o m e q u a t i o n (7) we deduce the following formu!a for the range R~ g Ucosat g + c V sin a T h e value of c depends on the initial velocity, the n a t u r e of t h e projectile, and the angle of elevation at which it is thrown. For each class of projectiles, and for each initial velocity, t a k e n at intervals of from 50 to IOO feet per second, t h e r e will be a certain curve whose abscissas will r e p r e s e n t the angles of elevation, and its ordinates t h e c o r r e s p o n d i n g values of c. Fz~. z shows such a curve, w h i c h gives the values of c in the e x p e r i m e n t s m a d e by the Chief of O r d n a n c e on the firing of two guns, and recorded in his report for i885, pp. 66, 495. One of the g u n s was a fifteen-inch R o d m a n s m o o t h bore. T h e projectile was r o u n d iron solid shot, and its w e i g h t 455 pounds. T h e initial velocity was 1,678 f. s. T h e other g u n was a 3"2 inch breech-loading rifle. T h e projectile was a solid b a n d e x p e r i m e n t a l shell, and its w e i g h t t h i r t e e n pounds. T h e initial velocity was 1,6o2 f. s. T h e values of c, p l o t t e d from these two sets of experiments, g a v e two curves, w h i c h ran so n e a r l y together, t h a t it was t h o u g h t best to combine t h e m into one m e a n curve, prov i d i n g t h a t the c o m p u t e d results w o u l d agree w i t h t h e experiments, as well as the e x p e r i m e n t s agree w i t h themselves.
Oct.,
I891.]
2DOf/'/Z o f
a Projectile.
271
T h e following table gives a synopsis of the results : ~5-INCH RODMAN
A n g l e of
V a l u e of c t a k e n frc, m
Elevation.
Diagrc m,
Fig. z,
GUN.
RAl~c-~
T I M g oF FLIGHT
" i! I By E x p e r i m e n t . l B y F o r m u l a . B y Experiment.: B y F o r m u l a .
fetl.
feet.
seconds.
• 2o
0'0673
5,300
5,23 °
4'0
5°
O'IZl2
8,857
8,8*0
8'i
7"9
Io 0
0'0890
73,696
13,636
I3" 9
r4' 9
i5 °
o'o748
16,65 I
I7,3z9
i9"i
2I' 5
~oo
o'o66o
x9,285
20,043
23"8
~7'7
23°
o"o622
21,243
2~,268
26"4
3*'2
3'2-1NCH BREECH-LOADING
Angle of Elevation.
2o
V a l u e of c t a k e n from Diagram,
F i g . I.
o'o673
I I [
seconds. 3"5
RIFLE.
RANOE
TIME
OF
FLIGIIT
]By E x p e r i m e n t .
fly F o r m u l a .
By E x p e r i m e n t .
By F o r m u l a .
feet.
feel. 4,797
seconds.
second. 3"3
6'7
4,755
4'o
6"2
4°
o'*I48
7,o93
7,096
6o
o'Io65
9,xo9
9,215
9'o
9'o
8°
0'0974
zo,9o7
ii,o66
Ii' 5
II" 7
io o
0"0890
x2.45o
I2,77o
13' 7
14"3
12°
o'e818
I4,472
I4,354
15' 7
x6' 9
o'o77 o
z5,8io
I5,658
x7' 9
19"4
16°
0"0725
I7~O7O
~6,89o
2o
ex" 9
18°
0"0690
x8,29~
I7,944
22
~4".3
2oo
0'066o
I9,437
I8,87o
24
26"6
14°
.
T h e fourth and fifth of the R o d m a n experiments s h o w the greatest discrepancies, a m o u n t i n g to about four per cent. of the range. It will be found, however, on e x a m i n i n g the separate experiments of which these are the mean, that they show differences a m o n g t h e m s e l v e s of from seven to eleven per cent. T h e initial velocity was m e a s u r e d only in the first few experiments, and it was a s s u m e d to be the same in all the others, as the w e i g h t and kind of powder was the same. But it is probable that this a s s u m p t i o n is
272
Gilman :
[J. F. I.,
n o t s t r i c t l y correct and it w oul d h a v e b e e n desi rabl e to h a v e h a d a m e a s u r e m e n t of t h e initial v e l o c i t y at each e x p e r i m e n t , as it seems likely t h a t t h e differences in t h e r a n g e s c o r r e s p o n d i n g to t h e s a m e a ngl e of e l e v a t i o n are p r i n c i p a l l y due to differences in t h e initial velocities. T h e c c u r v e g i v e n in Y@. z of a c c o m p a n y i n g plate, is pecul i ar in t h a t it shows s uch a g r e a t v a r i a t i o n in t he v a l u e of c f o r angles of e l e v a t i o n b e t w e e n 2 ° and 4 ° . B e y o n d 5°, as the a ngl e of e l e v a t i o n increases, t h e v a l u e of c g r a d u a l l y d i m i ni s he s , a n d t h e c u r v e is v e r y r e g u l a r . T h e c u r v e was c o n s t r u c t e d so as to gi ve as c o r r e c t v a l u e s as possible for t h e ranges, and w i t h o u t r e g a r d to t he times, w h i c h wer e left to c om e out as t h e y would. It will b e n o t i c e d t h a t th e c o m p u t e d t i m e s are s m a l l e r t h a n t h e o b s e r v e d for angles of e l e v a t i o n of f r o m 2 ° to 5 ° , t h a t f r o m 5 ° to 6 ° t h e o b s e r v e d and c o m p u t e d t i m e s are p r a c t i c a l l y t h e same, while b e y o n d 6 ° t h e excess of t he c o m p u t e d o v e r t h e o b s e r v e d v al ues g r a d u a l l y i ncr e a s es w i t h t he angl e of elevation. If it were r e q u i r e d to c o m p u t e t he t i m e w i t h g r e a t e r exactness, let t' d e n o t e t h e c o r r e c t e d value, t the v a l u e o b t a i n e d d ir ectl y f r o m f o r m u l a (5), and d e n o t i n g b y n t h e a n g le of e l e v a t i o n ( e xpr es s ed as seconds), we will h a v e f o r t h e R o d m a n gun, t ' = t--(Tn~) 2 and for t h e 3"2:inch rifle, t' =
t - - ~g.
T h e s e f o r m u l a s will a p p l y w h e n t h e a ngl e of e l e v a t i o n e x ceed s 8 °. F i g . ~ shows t h e c c u r v e for e x p e r i m e n t s m a d e w i t h t h e forty-two m i l l i m e t r e H o t e h k i s s m o u n t a i n gun, and r e c o r d e d in th e O r d n a n c e R e p o r t for I885, p. 88. T h e initial v e l o c i t y was a b o u t 1,3oo f. s., and the w e i g h t of t h e projectile, one p o u n d fifteen ounces. F i g . 3 r e p r e s e n t s t h e p a t h of a p r o j e c t i l e as d e d u c e d fr o m d a t a of one of t he R o d m a n g u n e x p e r i m e n t s a b o v e r e c o r d ed , t h e angl e of e l e v a t i o n b e i n g 23 ° , and t h e val ue of c o'o622. T h e v a l u e s of t h e o r d i n a t e s w e r e c a l c u l a t e d f r o m e q u a t i o n (7) fbr e v e r y 5oo f e e t of h o r i z o n t a l distance, and the r e s u l t s i~lotted, give t h e curve. It is u n s y m m e t r i c a l in form, and flatter on t he a s c e n d i n g t h a n on t he d e s c e n d i n g side, as is known_ by o b s e r v a t i o n to be t he case, in t h e
Oct., 189I.]
Pat,lz of a Projectile.
~73
a c t u a l p a t h of a projectile. T h e horizontal distance from t h e origin at w h i c h the projectile a t t a i n s its g r e a t e s t elevation is, in this example, a b o u t two-thirds of the range. In order to show how well these ordnance e x p e r i m e n t s s u s t a i n the h y p o t h e s i s of resistance proportional to the square of the velocity, we wili take the equations t h a t apply to this case, and which are g e n e r a l l y given in treatises on mechanics. t
--
I
(6 c r -
I)
6 [/r60S 8 62 ~1 6 0 S 2 ~'t" l a * l (~ -~2 g2cr _ _
2 67" - -
I
In these equations t represents the t i m e of flight, a the angle of elevation, z, the initial velocity, r the range, h tile h e i g h t due to the initial velocity, and c the c o n s t a n t of resistance. Since these equations can be applied only to cases in w h i c h the angle of elevation is v e r y small, we will select for comparison the first two of the R o d m a n g u n experiments. W e find t h a t the value of c, which is o b t a i n e d by substit u t i n g for r the correct range, is in the first e x p e r i m e n t o'oooo386, and in the second o'oooo8io. T h e corresponding realness of the times are 3"5 seconds and 7"7 seconds. W e see t h e n t h a t for angles of elevation b e t w e e n 2 ° and 5 o c is even more v a r i a b l e t h a n was f o u n d to be the case on the h y p o t h e s i s of a resistance proportional to the velocity, .'rod as the constancy of the value of c (at least for small angles of elevation), would seem to be a good test of the accuracy of the h y p o t h e s i s on w h i c h it is based, it follows t h a t the a s s u m p t i o n of a resistance proportional to the square of the velocity is no b e t t e r s u s t a i n e d by these'experiments, t h a n was the a s s u m p t i o n of resistance directly as the velocity. T h e values of c for the first two e x p e r i m e n t s w i t h the 3"2 inch breech-loading rifle, are 0"ooo0486 and o'oooo355 , and the c o r r e s p o n d i n g values of the times 3"3 and 6"t. It appears t h a t the resistance of the air can h a r d l y be expressed in t e r m s of a n y s i n g l e power of the velocity, b u t VOL. CXXXII. x8
274
Gillnan.
[J. F. I.,
I t h i n k it can be expressed quite a c c u r a t e l y by the following formula : R =
a + b y -t- c v ~
in which 1~ is the resistance, v the velocity, and a, b and c three constants to be d e t e r m i n e d by experiment. I h a v e applied this f o r m u l a to the e x p e r i m e n t s of H u t t o n on air resistance, and after d e t e r m i n i n g the values of the c o n s t a n t s b y least squares, f o u n d t h a t it satisfied those e x p e r i m e n t s w i t h i n the limits of probable errors of observation. T h e r a n g e of the velocities was from 95o to 2,o5o feet per second. T h e g r e a t e s t difference b e t w e e n the c o m p u t e d and experim e n t a l resistance was a b o u t three per cent., and the m o s t of t h e m were m u c h smaller. But, a l t h o u g h this f o r m u l a p r o b a b l y represents the law of resistance, y e t to a t t e m p t to m a k e it the basis of a m a t h e m a t i c a l d e d u c t i o n for finding the e q u a t i o n of a projectile's p a t h w o u l d be a hopeless problem, from its complexity. It seems more practicable to find the e l e m e n t s of the trajec_ t o r y by the first m e t h o d shown above; especially since the r e s u l t s of t h a t m e t h o d are as accurate as those arrived at on t h e g e n e r a l l y accepted h y p o t h e s i s of resistance propor tional to the square of the velocity.