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157. Therefore it is also true, when a force is applied to each of a system of particles, such that the magnitude of each force is proportional to the mass of the particle to which it is applied (so that each particle has produced in it a certain acceleration).

For example, the weight of any particle produces in it a certain fixed acceleration.

Example. A smail shell explodes in mid-air, in such a way that each fragment has the same additional initial velocity u velos away from the centre of the shell, given to it at the instant of explosion ; shew that (neglecting the resistance of the air) after any interval t secs. the frag: ments all lie in the circumference of a sphere of radius ut ft.

Each fragment is moving (i) with the velocity of the shell, say v velos at the instant of the explosion, (ii) with the additional velocity u velos, and (iii) with the acceleration g celos due to gravity, downwards.

To find its position after 't seconds we must draw from the initial position 0 (which is that of the shell at the instant of explosion), a line vt ft. parallel to v, a line 1 gta vertically downwards, arriving at a point O'; then from O' a line ut in a different fixed direction for each fragment.

The point O' is the same for each fragment; so that after t seconds each fragment is at a distance .ut ft. from the same point O'. This point O' is the position which the small shell would have occupied had it not exploded.

EXAMPLES. XXXVII.

1. A shell when moving horizontally explodes so that its mass is equally divided and one part is brought momentarily to rest by the explosion; shew that the velocity of the other part is doubled by the explosion; and if the shell was at the height of 400 ft. and moving with 100 velos, shew that the distance between the two portions on reaching the ground is 1000 feet.

2. A number of guns are fired simultaneously in different directions each making the same angle with the horizon, the muzzle velocity of the bullets being the same; prove that the bullets are after any interval to be found on the circumference of a horizontal circle,

3. A number of rifles are simultaneously discharged in all directions from the same point, the bullets having each a muzzle velocity of 1000 velos; prove that when the bullet which was projected vertically has reached its highest point, all the bullets are on the surface of a sphere of radius 31250 ft.

4. A number of particles are simultaneously swept off a table with different horizontal velocities; shew that they all reach the ground simultaneously.

5. A series of particles are simultaneously projected along a horizontal line on an inclined plane with different horizontal velocities; shew that as they slide down the plane they continue in the same horizontal line.

6. Shew that if a series of spheres are projected as in Question 5, which impinge on one another, they will always be in the same horizontal line.

7. Prove that the force which a railway engine exerts in pulling a train, over and above that necessary to overcome the friction, is distributed among the component parts of the train in the proportion of their

masses.

8. Shew that if any number of particles are moving with the same velocity in space, and parallel forces proportional to the masses of these particles are applied to each of them respectively, the relative positions of the particles will not be altered thereby.

CHAPTER XIII.

THE HODOGRAPH.

158. We represent the path of a moving point P by a line drawn on the paper.

The place from which the point is observed is represented on the paper by a fixed point 0; the distance of P from O is represented by the straight line OP.

Consider the positions P, P, which P has at the beginning and at the end of an interval t secs.

The additional distance of P in the interval t secs. is represented by the straight line P.P.

When P is moving with uniform velocity u velos, then PP, = ut ft.

159. When the path of P is a curved line, a tangent to the curve at a point Pu, gives the direction of the velocity of Pat the instant when the point is at P.

Take a point P, near to P.

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Then P.P, is a line whose direction approaches that of the tangent at Pı, when P, is made to move nearer to P.

Also, if P,P,=sft., and the point P take t secs. in going from P, to P, then approaches to the number of velos in the velocity of P when at P, as P, approaches P.

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160. Now take another fixed point Oʻ; from O' draw a line O'Q to represent the velocity of the moving point P. If we consider Q to move so that the line 0 Q is always parallel and proportional to the velocity of P, Q will trace out a line. The line traced in this way by the point Q is called the Hodograph of the point P.

161. On the hodograph take the two points lv le which correspond to the points P, P, on the path of P. Then Q. Q, represents the additional velocity added to Pin the interval t secs.

For OQ2 is the velocity at P2, and it is the resultant of the two velocities OQ, and QiQ2. [Art. 99.]

162. Suppose , l, represent y velos, then, as P, approaches P., the ratio approaches the number of celos

t in the acceleration which P has when at P.

Also, the direction of the tangent to the hodograph at Qı, is the direction of the acceleration of P when at P:

In other words, the velocity of Q in the hodograph represents the acceleration of P on the curve.

Example i. The hodograph of a point which moves with uniform velocity is a point.

The line OQ is fixed.

Example ii. When a point is moving with a velocity which is uniformly increasing in a given direction, the hodograph is a straight line along which it moves with uniform velocity.

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L. D.

Example iii. When a point is moving with constant speed [See Art. 78] in the circumference of a circle, its hodograph is a circle; and the point Q moves along the hodograph with uniform speed.

For OQ represents the velocity of P, and it is of constant magnitude and its direction turns through equal angles in equal intervals.

Example iv. When a point moves with constant speed v velos along the circumference of a circle of radius r feet, it has a constant

v2 acceleration celos whose direction always points to the centre of the circle.

Let the point P be describing the circle PP, whose centre is A with constant speed.

Draw the hodograph of P; let this be the circle QQ, whose centre is O; then the acceleration of P is represented by the velocity of Q.

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Let Q1, Q2 be points on the hodograph corresponding to the points P1, P, on the path of P.

Then OQ, is parallel to the velocity of P at P, and is therefore perpendicular to AP1.

The velocity of Q at l, is perpendicular to OQ1, which is parallel to P A and is constant. Therefore the acceleration of P at P, is in the direction P A and is constant.

Now Q describes the circumference of its circle while P describes the circumference of its circle ; let the interval in which this is done be t secs.; let the velocity of Q be a velos.

Then, O'Q=v ft. and at=2TV, also,

vt=277,

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Q. E. D.

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