common velocity at the end of compression, (2) their velocities at the end of restitution supposed to be perfect. Ans. (1) 21 in the direction of a's motion; (2) both rebound, 88. A mass of 100 lbs., perfectly free to move, is struck by a body weighing 2lbs., and moving with a velocity of 1000 ft. per second, which penetrates the mass and remains in it:-(1) Why must this case be treated as one in which the mutual action ceases at the end of compression? (2) What is the velocity with which the mass begins to move? (3) What is the energy of the mass at the end of the impact? (4) If the body were on a rough horizontal plane, how far would it move before coming to rest, the coefficient of friction being 0.2? Ans. (2) 1931 ft. per second; (3) 19,608 units; (4) 30 ft. 89.-Give examples to illustrate the meaning of the term 'potential energy.' In the last question, how much of the energy of the shot has been converted into 'potential energy?' at the end of the impact? Ans. 980,392 units. N.B. In the answers to the above questions, and elsewhere in this book, 32 is taken as the numerical value of the accelerative effect of gravity, unless the contrary is expressly stated or manifestly implied; e.g., in obtaining the 3rd answer to Q. 83 g is taken to equal 32-1912, this being manifestly implied in the mention of the place.. CHAPTER VI. MOTION IN A CIRCLE. 115. Composition of velocities. We have hitherto considered only those cases in which the force acts on the body along the line in which the motion takes place. We are now to consider one or two simple cases in which the force acts along a line transverse to the direction of the motion. In other words, hitherto we have considered only cases in which the force changes the velocity of the motion. We are now to consider some cases in which the force changes the direction of the motion. To enable us to do this we must consider the following question : A certain point has two velocities simultaneously impressed on it in different directions; what are the direction and magnitude of the resultant velocity? For instance, a ship sails due north at the rate of 4 miles an hour, a man walks across its deck towards the west at the rate of 4 miles an hour. The man has two velocities simultaneously impressed on him, viz. the velocity of the ship, and that due to his walking, and in consequence he is approaching the north and the west simultaneously with equal velocities, so that his actual motion is to the northwest. A little consideration will show that this result is generally true, so that if the one velocity is represented in magnitude and direction by A B, the other in magnitude and direction by a c, the resultant velocity will be represented in magnitude and direction by the diagonal A D of COMPOSITION AND RESOLUTION OF VELOCITIES. 161 A C FIG. 101. B the parallelogram, whose sides are A B and A c. On the other hand, a velocity represented in magnitude and direction by AD is equivalent to two coexistent velocities represented in magnitude and direction by the sides a B and AC of any parallelogram A B D C constructed on A D as a diagonal. AD Thus, if a train moves in a north-west direction with a velocity of 50 miles an hour, it approaches the north with a velocity a little exceeding 35 miles an hour, and simultaneously approaches the west with an equal velocity. On the whole, it will be seen that velocities admit of composition and resolution by the same rule as forces. 116. Applications. We shall now deduce two results from the last article which will be useful in certain questions which follow. (1) Let ABCD be any regular polygon inscribed in a circle whose centre is o. If A B is produced to H, so that в H and A B are equal, it is plain that HC is parallel to вo; if, then, Kc is drawn parallel to ви, виск is a parallelogram whose diagonal BC is equal to the side BH. body to move from ▲ Now suppose a velocity (v), and when it reaches B that another velocity (v) is impressed upon it in the direction Bo, such that VV::BK: BH. At the point в the body has simultaneously two velocities v and v, represented in magnitude and direction by BK and BH; consequently the resulting velocity is represented in magnitude and direction by BC; in FIG. 102. H B n K M other words, the consequence of the impressed velocity (v) is merely to change the direction of the motion, the velocity of the body being still v, but the motion taking place along BC. If when the body reaches ca velocity equal to v is again impressed on it along the line co, the body will describe the side CD with the velocity v. In a like manner it may be made to describe in succession all the sides of the polygon with the constant velocity v. M FIG. 103.. I H K P C M (2) Suppose a body to move with a uniform velocity in a circle ACBD. Through the centre o draw two diameters AB, CD at right angles to each other. It is required to determine the components of the velocity in directions parallel to A B and C D. Take P any point in the cir cumference; join P and o; draw PN and P M at right angles to AB and CD respectively. Draw PH at right angles to Po, and equal to it; complete the rectangle P K HL; PL is in BNP produced, and it is plain that PK is equal to PN, and PL to PM. Now as PH touches the circle at P, the body when at P is moving in the direction PH; consequently PH may be taken to represent the velocity; and, if this is done, PK and PL represent the component velocities required.. We see, therefore, that when the body is at any point P, the velocity parallel to AB is to the velocity in the circle as the perpendicular line PN is to the radius Po. D FIG. 104. A result of some importance can easily be deduced from this. Take AB, any line whose middle point is o, and on AB as a diameter describe a circle; take any point P in the circumference, and draw PN at right angles to A B, and join P and 0. Suppose two bodies to set out from A, the one moving along the circumference with a uniform velocity v, the other moving along the diameter with a variable velocity (v), such that at any point (N) B N VVPN: PO. Under these circumstances the bodies will reach B together; for at each instant they are moving with equal velocities in the direction ▲ B. Also since the former body describes the semicircle with a uniform velocity V, the time occupied by the bodies in reaching B will equal ▲ B÷v. 117. Uniform motion in a circle.-When a point whose mass is м moves with a uniform velocity (v) in a circle whose radius is r,. it must be acted on by a force P, given by the equation M V2 This force acts along the radius towards the centre. if more forces than one act on the body, their resultart UNIFORM MOTION IN A CIRCLE. 163 will be equal to the force r, and will act as stated, viz. along the radius and towards the centre. In the above formula, if м is in pounds, r in feet, and v in feet per second, P will be in absolute units. Ex. 133.-A mass of 12 lbs. is tied to the end of a string 8 ft. long, and is whirled round in a circle at the rate of 10 ft. per second; P will equal 12 × 1028, or 150 absolute urits, equal to about 4.6875 lbs. And this force must be exerted along the string towards the centre. 118. We will now consider the reason of the statements made in the last article, and, first, that the force must always be directed towards the centre.. We have already seen (Art. 116) that if a point describes the sides of a regular polygon with a uniform velocity, there must be impressed upon it at each angle a certain velocity in the direction of the radius towards the centre; this velocity must be communicated by a force acting intermittently in the direction of the velocity, i. e. along the radius towards the centre. Now this result is true, however great be the number of sides, and therefore will be true of the limiting case to which the case approaches when the number of sides becomes very great, and the intervals between the successive exertions of the force very small. In this case we have the body moving in a circle with a uniform velocity, and force acts continuously along the radius towards the centre. Secondly, suppose the force to be one of P units, then if P=Mf, the velocity which P will communicate to the body in a time t is ft. In fig. 102 draw cn at right angles to в M; then we know from geometry that B C2 BMX Bî. Suppose the body to describe one side of the polygon in a time t, then BC=vt. Also, if we suppose the velocity ft to be suddenly impressed on the body at B, we have B.K, |