by a string passing over a small pully at the top of the plane: if the acceleration be one-fourth of that of a body falling freely, find the ratio of Q to P. 6. Two weights P and Q are connected by a string; and hanging over the top of a smooth plane inclined at 30° to the horizon, can draw P up the length of the plane in just half the time that P would take to draw up Q: shew that Q is half as heavy again as P. 7. Four equal weights are fastened to a string: find how they must be arranged so that when the string is laid over a fixed smooth pully, the motion may be the same as that produced when two of the weights are drawn over a smooth horizontal table by the weight of the other two hanging over the edge of the table. 8. Two weights of 5 lbs. and 4 lbs. together pull one of 7 lbs. over a smooth fixed pully, by means of a connecting string; and after descending through a given space the 4 lbs. weight is detached and taken away without interrupting the motion: find through what space the remaining 5 lbs. weight will descend. 9. Two weights are attached to the extremities of a string which is hung over a smooth pully, and the weights are observed to move through 64 feet in one second; the motion is then stopped, and a weight of 5 lbs. is added to the smaller weight, which then descends through the same space as it ascended before in the same time: determine the original weights. 10. Find what weight must be added to the smaller weight in Art. 89, so that the acceleration of the system may have the same numerical value as before, but may be in the opposite direction. 11. Solve the problem in Art. 92, supposing the inclined planes rough. 12. If the pully in Art. 89 can bear only half the sum of the weights of the two bodies, shew that the weight of the heavier body must not be less than (3+2√2) times the weight of the lighter body. IX. The Direct Collision of Bodies. 93. We have hitherto spoken of force as measured by the momentum which it generates in a given time; and the force with which we are most familiar is that of gravity, which takes an appreciable time to generate in any body a moderate velocity. There are however examples of forces which generate or destroy a large velocity in a time which is too brief to be appreciated. For example, when a cricket ball is driven back by a blow from a bat, the original velocity of the ball is destroyed, and a new velocity generated; and the whole time of the action of the bat on the ball is extremely brief. Similarly when a bullet is discharged from a gun, a large velocity is generated in an extremely brief time. Forces which produce such effects as these are called impulsive forces, and the following is the usual definition: An impulsive force is a force which produces a finite change of motion in an indefinitely brief time. 94. Thus an impulsive force does not differ in kind from other forces, but only in degree: an impulsive force is a force which acts with great intensity during a very brief time. As the laws of motion may be taken to be true whatever may be the intensity of the forces which produce or change the motion, we can apply these laws to impulsive forces. But since the duration of the action of an impulsive force is too brief to be appreciated, we cannot measure the force by the momentum generated in any given time: it is usual to state that an impulsive force is measured by the whole momentum which it generates. 95. We shall not have to consider the simultaneous operation of ordinary forces and impulsive forces for the following reason: the impulsive forces are so much more intense than the ordinary forces, that during the brief time of simultaneous operation, an ordinary force does not produce an effect comparable in amount with that produced by an impulsive force. Thus, to make a supposition which is not extravagant, an impulsive force might generate a velocity of 1000 in less time than one-tenth of a second, while gravity in one-tenth of a second would generate a velocity of about 3. 96. The student might perhaps anticipate that difficulties would arise in the discussion of questions relating to impulsive forces, but it will appear as we proceed that the cases which we have to consider are sufficiently simple. We may observe that the words impact and impulse are often used as abbreviations for impulsive action. 97. We are about to solve some problems relating to the collision of two bodies; the bodies may be considered to be small spheres of uniform density, and, as before, we take no account of any possible rotation: see Art. 10. The collision of spheres is called direct when at the instant of contact the centres of the spheres are moving in the straight line in which the impulse takes place; the collision of spheres is called oblique when this condition is not fulfilled. 98. When one body impinges directly on another, the following is considered to be the nature of the mutual action. The whole duration of the impact is divided into two parts. During the first part a certain impulsive force acts in opposite directions on the two bodies, of such an amount as to render the velocities equal. During the second part another impulsive force acts on each body in the same direction respectively as before, and the magnitude of this second impulsive force bears to that of the former a ratio which is constant for any given pair of substances. This ratio lies between the limits zero and unity, both inclusive. When the ratio is unity the bodies are called perfectly elastic; when the ratio is greater than zero and less than unity the bodies are called imperfectly elastic; and when the ratio is zero the bodies are called inelastic. The ratio is called the modulus of elasticity, or the coefficient of elasticity, or the index of elasticity. 99. There are three assumptions involved in the preceding Article, We assume that there is an epoch at which the velocities of the two bodies are equal; this will probably be admitted as nearly self-evident. We assume that during the two parts into which the whole duration of the impact is divided by this epoch, the action on one body is equal and opposite to the action on the other; this is justified by the Third Law of Motion. We assume that the action on each body after the epoch is in the same direction as before, and bears a certain constant ratio to it; this assumption may be taken for the present as an hypothesis, which is to be established by comparing the results to which it leads with observation and experiment. See Art. 104. 100. We have still to explain why the words elastic and inelastic are used in Art. 98. It appears from experiment that bodies are compressible in various degrees, and recover more or less their original forms after the compression has been withdrawn: this property is termed elasticity. When one body impinges on another, we may naturally suppose that the surfaces near the point of contact are compressed during the first part of the impact, and that they recover more or less their original forms during the second part of the impact. 101. A body impinges directly on another: required to determine the velocities after impact, the elasticity being imperfect. Let a body whose mass is m, moving with a velocity u, impinge directly on another body whose mass is m', moving with a velocity u. Let R denote the impulsive force which during the first part of the impact acts on each body in opposite directions. Then at the end of the first part of the impact, the momentum of the body of mass m ти-R is mu-R, and therefore its velocity is m : and the momentum of the body of mass m' is m'u' + R, and thereThese velocities are equal fore its velocity is m'u' + R m' Let e denote the index of elasticity; then during the second part of the impact an impulsive force eR acts on each body in the same direction respectively as before. Let denote the final velocity of the body of mass m, and that of the body of mass m'; then mu−(1+e)R (1+e) m' (u — u') v= =u m 102. From the general formula of the preceding Article many particular results may be deduced; we will give some examples. If the bodies are perfectly elastic, e=1; then we have |