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68. These equal and opposite impulses produce equal and opposite mass-velocities, one in one mass, the other in the other. Hence, the algebraic sum of the additional mass velocities produced during an impact is zero; in other words,

when two masses impinge their total massvelocity is unaltered by the impact.

69. Two masses M and M' are said to be inelastic when after direct impact they move each with the same velocity.

NOTE. When one mass sticks to another on impact; or when one mass penetrates another, as a bullet penetrates a piece of wood, and stays in it; in fact in every case in which the masses have the same velocity after impact, they are said to be inelastic.

Example i. A mass M of 3 lbs. having 5 velos impinges directly on a mass M' of 7 lbs. at rest; find their subsequent motion, the masses being inelastic.

Since the masses are inelastic, their velocities are made equal by the impact; let their velocity after impact be v velos. Before impact

sthe mass-velocity of Mis 3 x 5 pound-velos.

the mass-velocity of M' is o. After impact

sthe mass-velocity of M is 3 v pound-velos.

(the mass-velocity of M' is 7 xv pound-velos. Hence, by Art. 68, 3v + 70=15; or,

v=; that is, after impact they move together with a velocity of 13 st. per second.

Example ii. A leaden ball of 1 lb. impinges directly with a velocity of 1000 ft. per sec. on a mass of 1 cwt. moving in the same direction with a velocity of 20 ft. per sec; find their subsequent velocity.

Let them have v velos after the impact; then, since the total massvelocity is unchanged

1 X 1000 + 112 X 20=vx (112+1),

v=331439

= 28.6... That is, the masses move together with a velocity of 28•6... ft. per second

or,

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EXAMPLES. XIX.

The following impacts are all direct. 1. An inelastic ball of 1 cwt. moving with a velocity of 20 ft. per sec. impinges against an equal ball at rest; find their subsequent velocity.

2. An inelastic particle of m lbs. having v velos impinges directly on another of m' lbs. at rest; find their subsequent velocity.

3. An inelastic particle of 20 lbs. meets another of 2 lbs.; each particle has ni velos but in opposite directions; find their subsequent motion.

4. A railway carriage of 10 tons is moving towards a train of 100 tons with a velocity of 4 miles an hour; at the instant of collision the carriage is coupled to the train ; find the subsequent velocity.

5. A particle of 1 cwt. meets another of 28 lbs. having a velocity of 15 miles an hour and after the impact they both are at rest ; with what velocity was the i cwt. moving ?

6. A particle of 1 lb. moving with 30 velos overtakes another particle of 2 lbs. and after impact they move together with 25 velos; what was the velocity of the 2 lbs. before impact ?

7. An inelastic mass impinges on another of twice its mass at rest; shew that the impinging body loses two-thirds of its velocity by the impact.

8. An inelastic particle of mass m and velocity v impinges directly on a particle of mass m'; they are at rest after the collision; shew that

1 the velocity of the second particle was

m' 9. Three equal inelastic balls are placed in a line, not in contact ; an equal ball moving in the same line with 20 velos impinges on the first, these two impinge on the second and then the three on the third ; find their joint velocity after impact.

10. An engine of m tons impinges with a velocity of hour on each of 4 inelastic trucks each of m tons placed on the same line of rails and separated by a small interval; find the velocity of the train after impact.

11. A rifle bullet of } oz. having 1200 velos impinges on a block of wood weighing 5 cwt. at rest and free to move in the direction of the impact; what is the velocity of the block immediately after impact.

12. A particle is let fall from the top of a tower and at the same instant an equal particle is thrown vertically upwards from the foot of the tower with a just sufficient velocity to carry the particle under the action of gravity to the top of the tower. The two particles being inelastic impinge ; shew that directly after impact the particles are at rest; and if the tower is 128 ft. high find how long the first particle takes to reach the ground.

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ELASTICITY. 70. DEF. Two masses are said to be elastic when after impact, their velocities in the direction of the impulse are not the same.

When two elastic masses impinge they probably behave much as inelastic masses up to the time when they are moving with the same velocity ; the property of elasticity seems to cause a continuation of the impulse so that the masses recoil from each other. The amount of recoil must be in some degree proportional to the magnitude of the impulse and will depend on the nature of the material of the masses.

71. Newton observed, that when two spheres of given substance impinge directly on each other, the relative velocity after impact is e times the relative velocity before impact and in the opposite direction; where e is a number not greater than 1. This number e depends on the nature of the substances of the two spheres, and is called their coefficient of restitution; hence we say that

72. When two elastic masses m lbs. and m' lbs. impinge directly on each other, u velos and u velos being their velocities before impact, v velos and ' velos their velocities after impact, then

v v = -e(u u'); where e is a number, not greater than 1, which depends on the materials of the masses and is called the coefficient of restitution.

In other words, the two masses separate after impact with a velocity which is e times that velocity with which they were approaching each other before impact.

Velocity of separation=e times velocity of approach.

When the coefficient of restitution of a substance = I the substance is said to be perfectly elastic.

(ii)

Example i. Two masses of 5 lbs. and 20 lbs. having velocities of 20 velos and

10 velos, whose coefficient of restitution is , impinge ; find their subsequent velocities.

Let v velos and v' velos respectively be their subsequent velocities.
Then since their total mass-velocity is unaltered by the impact
5 X 20 - 20 X 10=50+ 200'

(i) also, by Art. 41, v-v=- }(20+10) whence

50+ 20v' = - 100, and

W-U= - 20; .. 250= - 100 – 400 ; or, v= - 20; also, d=0. Hence, after the impact the 5 lbs. moves in the negative direction with velocity 20 ft. per second, and the 20 lbs. comes to rest.

Example ii. Two masses of m lbs. and m' lbs. impinge with velocities u velos and u' velos; find the condition that they may interchange velocities.

Here u' velos and u velos are to be their respective velocities aster impact; hence, mu + m'u'=mu' + m'u

(i) or,

(m - m) (x - 4')=o, whence,

m=m (for, since the masses impinge, u cannot =u'). Again, 4- 4'= e (4 4)

(ii) or,

(u - U') (1 - e)=0, whence,

Thus, in order that two masses after impact may interchange velocities they must be of equal mass and perfectly elastic.

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e=I.

EXAMPLES. XX.

The coefficient of restitution is here denoted by e.

1. A sphere of 6 lbs. having 20 velos overtakes another of 4 lbs. having 12 velos; determine their velocities after impact, in the case in which e=i.

2. A particle of 28 lbs. having 8 velos impinges directly on another of 14 lbs. having – 16 velos ; determine their subsequent velocities when e=1.

3. Two elastic balls (e=!) of masses 3m and m and velocities 3v and – 5v impinge ; find their subsequent velocities.

4. A ball of 3m impinges on another m at rest which afterwards moves with 10 velos, ę being ; find the velocity of the first ball before impact.

5. The result of a direct impact between two spheres of elasticity 1, one of which is at rest, is that one of them has after impact twice the elocity of the other; prove that one ball has the mass of the other.

6. A series of equal elastic balls of elasticity e are placed in a line separated from each other by short distances; another ball of equal mass moving in the same line impinges on the first with velocity v; find the velocities after impact of the first, second, third and nth balls.

7. A ball of mass m and elasticity e is projected vertically under the action of gravity with 64 velos; another ball of mass m is simultaneously let fall from a height of 64 ft. vertically above m; find when and where they impinge, and their velocities after the impact.

8. Two elastic spheres impinge directly with equal velocities; find the ratio of their masses that one of them may be reduced to rest by the impact.

9. A shot of 1 cwt. is projected from a 20 ton gun placed on a smooth horizontal plane, with a horizontal muzzle velocity of 3000 velos; find the velocity of the recoil of the gun.

10. Two railway carriages B and C of m lbs. and m' lbs. stand on the same line of perfectly smooth rails separated by a short distance ; a third carriage A of m lbs. impinges on B and then consequently B impinges on C; prove that A will impinge a second time on B if m' is

where e is the coefficient of restitution.

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11. Two equal spheres A, B are connected by a string and laid on a smooth horizontal table, at a distance from each other which is less than the length (l ft.) of the string; a velocity u is given to A in the direction ; on the string becoming tightened a direct impulsive tension is set up; if the coefficient of restitution of the string be e, and of the spheres themselves é, find the velocity of the spheres (i) after the first impulse of the tension, (ii) after the first impact of the spheres.

12. A series of equal masses of 1 lb. each are connected by light inelastic strings each

ft. long. They are placed in a straight row touching each other on a smooth horizontal plane; to the first mass is applied a force of 2 poundals in the direction of the row, and when the first mass has moved over 4 ft., the string applies a direct impulse to the second mass; and when the first two have moved over 4 ft. more, the string applies a direct impulse to the third mass, and so on; find the initial velocities of the second, third and fourth masses.

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