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resistance of the air, and this is so considerable as to render the conclusions drawn from the theory almost entirely inapplicable in practice. From experiments made to determine the motion of cannon-balls, it appears that when the initial velocity is considerable, the resistance of the air is 20 or 30 times as great as the weight of the ball; and the horizontal range is often a small fraction of that which the preceding theory gives. Such experiments have been made with great care, and shew how little the parabolic theory is to be depended upon in determining the motions of military projectiles.

From a long series of experiments made at Woolwich, Dr Hutton arrived at the conclusion that the velocity v of a cannon-ball on quitting the gun could be nearly expressed by 2P the formula v = 1600 W

Pbeing the weight of the charge

of powder and W that of the ball.

And further, if the projectile be of finite size, and have a rotatory as well as a progressive motion, the resistance of the air, which acts along the surface of the body (or tangentially), will in general change its direction, or the plane of its motion, or both. For this resistance increases with the velocity and the density of the air, and will consequently be greater on that side of the body where the rotatory and progressive motions conspire, than on the other side where they oppose each other: and the density of the air immediately in front of the body is greater than behind it.

Another cause of irregularity will also exist if the ball be not homogeneous-as for example if it contain air-bubbles within, from imperfection in the casting-so that its centre of gravity does not coincide with its centre of figure.

The non-symmetrical action of these causes on the body. will make it deviate from its plane of motion, except in the single case when the axis of rotation coincides with the direction of progressive motion. On this principle has been explained the irregular motion of a tennis-ball and the deviation of a bullet from the vertical plane. It is in a great measure remedied in the case of a rifle ball, since the rifling of the barrel communicates to the ball a rotation about an axis in the direction in which the ball is projected.

(See Robins' Gunnery; Hutton's Tracts; Art. Gunnery in the Encyclopædia Britannica.)

CHAPTER V.

MOTION ON A CURVE.

95. WHEN a body moves along a smooth curve the curve exerts a pressure or reaction upon the body at every point, but since this reaction is always perpendicular to the curve, it has no tendency to accelerate or retard the body. In order to determine the velocity of the body in any position we must resolve the forces upon the body in direction of the motion at successive instants, and examine the effect of these resolved forces.

96. An inelastic particle descends down a smooth curve in a vertical plane under the action of gravity, to find the velocity of the particle in any position.

We may regard the curve as the limit of a polygon whose sides are equally inclined to one another, by supposing the number of sides to be indefinitely increased, and the angle between consecutive ones to become evanescent.

Let AA....A be such a polygon; draw Aa, A,α, Д,a,... perpendiculars on the vertical line through A.

Let be the angle between successive sides of the polygon which are not necessarily of equal length,

An

A1a,

A

Aa

a3

A4

a4

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2

2

2

AA,,

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when

the particle

comes to A, it is deflected in direction AД, and starts along AA, with velocity

2

v1 cos 0;

'

v„2 + (v22 + v ̧2 + ... + v3n-1) sin2 0 = u2 + 2g. Aα...... (i).

Now if a be the angle between the directions of motion at A and A„, and v' the greatest of the velocities v1, V2...

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and (v2+v+ ... + v3n-1) sin2 0 < (n − 1) v2 sin2 0 ;

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and this vanishes in the limit when n is indefinitely increased, a remaining finite, (in which case the polygon becomes the curve);

2

2 2

:. à fortiori (v2+v+...+v2) sin2

vanishes in the limit, in comparison with v.

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Hence in the limit when the polygon becomes the curve, the equation (i) becomes

2

vn2=u2 + 2gh,

which expresses the velocity at any point on the curve in terms of the initial velocity, and the vertical height through which the particle has fallen; or suppressing the suffix,

v2 = u2+2gh.

97. Obs. In the above investigation we have supposed the particle inelastic and moving on the concave side of the curve, towards which the force of gravity pulls the particle; these suppositions being made in order that the particle may remain in contact with the curve. We shall see hereafter that a particle moving on a curve will, under certain conditions, quit the curve; but the necessity of the supposition here referred to would be obviated by supposing the polygon AД,4,... to be a polygonal tube (becoming a curvilinear one in the limit) of small bore, just sufficient to allow the free passage of the particle. The result arrived at for the velocity at any point would hold good in this case, and will be equally true for a particle moving either on the concave or convex side of a curve, so long as it remains in contact with the curve.

1

COR. 1. If the particle start from rest at A, then u=0, and v2 = 2gh; i.e. the velocity, acquired from rest, down a smooth curve is equal to that which would be acquired by a body falling freely through the same vertical height. More generally we may interpret the equation v2 = u2 + 2gh thus: the square of the velocity at any point A,, is equal to the square of the velocity at any other point A, increased by the square of the velocity which the force of gravity would generate in the body in drawing it from rest through the same vertical space.

This result it will be observed is independent of any particular form of the curve.

COR. 2. If a body be projected up a curve, the vertical

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