the ball A will be at rest after the impact;

and according as e<or >

13
15

A will follow B with a less

velocity or be reflected back and move in the opposite direction.

64. (II) A ball A moving with a given velocity impinges directly upon a ball B at rest, and B afterwards impinges directly upon a ball C at rest; find the velocity communicated to C.

If u be the original velocity of A, we have by Art. (56),

COR. 1. The velocity communicated to C by the intervention of B will vary with the magnitude of B, and will be the

this will be the case when B=√(AC);-in other words, the velocity of C will be greatest when B is a mean proportional between A and C.

65. (III) A particle is to be projected from a given point P so as to pass through another

given point Q, after being reflected at a given fixed plane AB; to find the direction of projection.

Suppose T to be the point where the particle must strike the plane, then the plane PTQ must

B

be perpendicular to the fixed plane, and will cut it in a straight line AB.

Now the particle impinging on the plane in direction PT and being reflected in direction TQ, we must have

tan QTS= e.tan PTA......(i) Art. (60).

If QS be drawn perpendicular to AB, and PT produced to meet QS in R, we shall have

tan QTS e tan RTS,

and therefore QS-e. SR.

=

This suggests the following simple construction for determining T. Draw QS perpendicular to AB and produce it to R, making SR=1. QS; join PR cutting AB in T. Then

e

the condition (i) is satisfied, and PT is the direction in which the particle must be projected.

COR. If the particle is to pass through Q after reflexion at two planes TV, US in succession, we have the following

construction. Draw QSR perpendicular to the latter plane, making

SR =1. QS.

Draw RVD perpendicular to the first plane, making

VD=1.RV;

e

Ꭱ

join PD cutting the first plane in T,-join TR cutting the second plane in U,—then if the particle be projected in direction PT it will be reflected along TU and again reflected at U in direction UQ, and so pass through the point Q.

66. (IV) A heavy particle impinges upon a fixed rough plane; to find its motion after impact.

Let the plane of the paper represent the plane of impact, i. e. the plane which contains the

direction of motion of the particle before impact, and the normal to the fixed plane at the point of

contact.

Let u, u' be the velocities of

the particle (mass A) before and after impact.

α, O the angles its direction of motion makes with the normal QN before and after impact.

X, F the momentum generated by the fixed plane in the particle, in directions QN and QC,-the latter arising from the roughness of the plane.

Then resolving the motion in direction QN and CD, we have, as in Art. (60),

u cos 0 =eu cos a......

the complete value of X= (1+e) Au cos a.

and Au' sin 0 = Au sin α - F..........

(i),

(iii).

Now we may take F=μX (iv), where μ depends upon the roughness of the plane, and is a numerical quantity to be determined by experiment, it is sometimes called the coefficient of dynamical friction; from these four equations we get

which two equations determine u' and 0, i. e. the velocity and direction of motion after impact.

CHAPTER III.

OF UNIFORMLY ACCELERATED MOTION.

67. THE accelerating force upon a particle is said to be uniform when equal increments of velocity are added in equal increments of time, however large or small these increments of time may be.

Hence, in accordance with the definitions and conventions of Arts. 5, 13, if v be the velocity of a particle at the end of a time t, during which it has been subject to a uniform accelerating force ƒ, and if u were its velocity to begin with, we shall have f. t to represent the increment of velocity, and v=u+ft....... ........(i).

If the particle started from rest u = 0 and v=ft.

Obs. The formula (i) is algebraically true in any case where the force is really a retarding force (Art. 6, Obs.), or where the velocity u at the beginning of the time t exists in a direction opposite to that in which is measured: in any case it is necessary simply to assign the proper algebraic sign to u and ƒ, and the result (i) will be available.

68. If s be the space described from rest in time t by a particle under the action of a uniform accelerating force f, then will s = ft.

Let the time t be subdivided into n intervals, each equal to τ, so that nτ= t; then the velocities at the beginning of the

1st 2nd 3rd

will be 0 fr 2ft...

nth of these intervals of T

(n-1) fr;