No. 6. Case 4. Let C fall from 12 inches, and D from 6 inches in the same direction. Before the stroke, the velocity of C is 12, and quantity of matter 2; whence its momentum is 24; q 2 × v 12 = m 24; and the velocity of D is 6, and its quantity of matter 1; whence q 1 × v 6 = m6; therefore the whole momentum is 30. After the stroke, the velocity of the whole is 10, and the quantity of matter 3; whence 93 X V10= m 30.

No. 7. Prop. XXI. Case 3. Let C fall from 6 inches, and D from 12, in opposite directions, the quantity of matter in C being 2, and its velocity 6; and the quantity of matter in D being 1, and its velocity 12, their momenta will be equal, and being opposite, will destroy each other. Cq2 x v6 = m 12; Dq1 x v 12 =m 12.

No. 8. Case 4. Let C fall from 3 inches, and D from 12, in opposite directions: Before the stroke, the momentum of C is 6; q 2 × v 3 = m 6, and the momentum of D is 12; q 1 × v 12 = m 12; whence the difference of their momenta is 6. After the stroke, the velocity is 2, and quantity of matter 3; whence the momentum is 6; 9 3 × v 2 = m 6.

PROP. XXII. When one elastick body strikes upon another of the same kind, the one loses, and the other gains, twice as much momentum, as if the bodies had been void of elasticity.

For, since (by Def. III.) perfectly elastick bodies, on percussion, restore themselves with a force equal to that with which they are compressed, whatever momentum is gained by one body, or lost by the other, on percussion, from the law of re-action, the same must be gained, or lost, from the power of elasticity.

I.

COR. 1. Hence the momentum of elastick bodies after percussion may be found, by doubling the momentum which would have remained, had the bodies been non-elastick.

2. If one of the bodies, considered as non-elastick, would lose more than half its momentum, as elastick, it loses more than all, that is, acquires a negative momentum in a contrary direction.

Exp. The following experiments may be made with ivory balls suspended from strings ; they correspond with the preceding experiments on non-elastick bodies.

Let A and B be equal balls; and let Ċ be a ball double of the ball D.

No. 1. A, falling from 18 inches on B at rest, has 18 degrees of momentum before the stroke: therefore, after the stroke, supposing the balls non-elastick, the same momentum belonging to the two equal balls together, each has 9 degrees of momentum, and A has lost and B gained 9. This being doubled, A, as elastick, will lose 18, and B will gain 18 degrees of momentum : whence A will be at rest, and B will move with 18 degrees of momentum.

No. 2. A, falling from 18 inches, and B from 9 in the same direction; as non elastick, after the stroke, each has 13 momentum, or A has lost 4, and B gained 41. As elastick, after the stroke, A loses 9, B gains 9; therefore A rises to 9 inches, B to 18.

No. 3. A and B, falling in opposite directions from 12 inches, as non-elastick, would lose all their momentum; as elastick, each loses 24 degrees of momentum; that is, gains 12 in the contrary direction.

No. 4. A, falling from 12 inches, and B in the opposite direction from 6, as non-elastick, the momentum of each, after the stroke, will be in the direction of A; whence A loses 9, and B loses 9, moving 3 degrees in the contrary direction. As elastick, A loses 18, or has 6

in the contrary direction, and B loses 18, or gains 12 in the contrary direction.

No. 5. C, double of D, falling from 12 inches on D at rest, the momentum of C, before the stroke, being 24, and of D nothing; as non-elastick, C, after the stroke, having its momentum 16, and moving with the velocity 8, will have lost 4 degrees of velocity, and 8 of momentum: and D will have gained 8 of each. As elastick, therefore, C will lose 8 degrees of velocity, or (Prop. XI.) 16 of momentum, and D will gain 16 of each; that is, C will move with 4 degrees of velocity, and D with 16.

No. 6. C, falling from 12 inches, and D from 6 in the same direction, before the stroke, the velocity of C is 12, and its momentum 24; and the momentum of D 6. After the stroke, as non-elastick, the momentum of C is 20, because q 2 × v 10 = m 20; and the momentum of D is 10, because ? I XV 10 = m 10; therefore C has lost 4 degrees of momentum, or 2 degrees of velocity, and D gained 4 of each. If, therefore, the gain or loss be doubled on account of the elasticity of C and D, C will lose 8 degrees of momentum, or 4 of velocity, and D will gain 8 of each; that is, C will move with 8 degrees of velocity, and D with 14.

No. 7. C, falling from 6 inches, and D from 12, in opposite directions, their momenta being equal, would destroy each other without elasticity: Therefore, being elastick, each will acquire the momentum of 12 in opposite directions; that is, D will return to 12, and C to 6.

No. 8. C, falling from 3 inches, and D from 12 in opposite directions; since the momentum of C, before the stroke, is 6, and of D 12, as non-elastick bodies they would, after the stroke, move in the direction of D, with the velocity of 2; whence C would move in the direction contrary to its first motion with 4 degrees of momentum, and lose 10; and D would lose 10: Therefore, being elastick, C will lose 20 degrees of momentum, and also D 20; whence C will move in the contrary direction with 14 degrees of momentum; that is, will return to 7; and D will return to 8.

COR. I. If the sum of two conspiring momenta, or the difference of two contrary momenta, be divided by the sum of the quantities of matter in both the moving bodies, the quotient will give the common velocity after the stroke.

SCHOL.

SCHOL. Let A and B be two spherical bodies, moving with their centres in the same line ; and let their velocities be a and b. The momentum of A, before the stroke is Aa, and that of B is Bb; their sum, or their difference, is Aa+Bb, or Aa-Bb. Therefore (by Prop. XX. and XXI.) the momentum, after the stroke, is expressed by Aa+ Bb, and, their common Aa+Bb AAa+ABb velocity by A+B ABa+BBa

of B is

A+B

Hence the momentum of A, after the stroke, is

A+B ; and that

Next, suppose the bodies perfectly elastick. Subtract the momentum of A considered
A Aa+ABb

as non elastick, after the stroke,

and the remainder,

A Ba+ABb

A+B will express the momentum in that case lost by A, and

tum of B, after the stroke, supposing them perfectly elastick. And

2 Aa+Bb+Ab

A+B , will express their respective velocities.

Aa+2 Bb-Ba
A+B

COR. 2. If there be any number of elastick, equal, and spherical bodies, whose centres are placed in the same line, and the first body strikes upon the second in the direction of that line, all the bodies will be at rest except the last, which will move off with the velocity of the first.

EXP. Several equal ivory balls, being so suspended as to have their centres in a right line, if the first be let fall upon the second, the last will fly off, to the height from which the first fell.

COR. 3. When the striking ball is less than the quiescent, there will be an increase of

momentum.

EXP. Let the ball D fall from 12 inches upon C, double of D, at rest. If they were non-elastick, they would proceed together, and their velocity, being the same, C, after the stroke, would have double the momentum of D; that is, C would have 8 degrees, and D whence

4;

23

whence D would have communicated more than half its momentum to C. The effect being doubled by the elasticity of the bodies, D communicates to C 16 degrees of momentum, and loses as much itself, or returns with 4 degrees of momentum in the contrary direction : while C moves forwards with 4 degrees more momentum than D had at the first. Thus the whole sum of momentum is increased from 12 to 20 degrees; but as much as the momentum is increased in the direction in which D first moved, so much is given to D in the contrary direction. In this manner may momentum be continually increased by a series of bodies.

COR. 4. If a non-elastick body strikes upon an immoveable obstacle, it will lose all its motion; an elastick body will return with a force equal to the stroke.

EXP. Let a leaden ball, and an ivory ball, strike upon any fixed plane.

CHAP. V.

Of Motion, as produced by the Attraction of Gravitation.

SECT. I.

Of the Laws of Gravitation in Bodies falling without Obstruction.

PROP. XXIII. The motion of a body, falling freely by the attraction of gravitation, is uniformly accelerated, or its velocity increases equally in equal times.

A new impression being made upon the falling body, at every instant, by the continued action of the attraction of gravity, and the effect of the former (by Prop. I.) still remaining, the velocity must continually increase. Suppose a single impulse of gravitation, in one instant, to give it one degree of velocity; if, after this, the force of gravitation were entirely suspended, the body would continue to move with that degree of velocity, without being accelerated or retarded. But, because the attraction of gravitation continues, it produces as great a velocity in the second instant as in the first; which being added to the first, makes the velocity in the second instant double of what it was in the first. In like manner, in the third instant, it will be tripled; quadrupled in the fourth and in every instant, one degree of velocity will be added to that which the body had before; that is, the motion will be uniformly accelerated.*

COR. The velocities of falling bodies, are as the times in which they are acquired.

PROP.

* All bodies descending in vacuo by gravity, whether great or small, dense or rare, are found to fall through 16.1 feet in one second, and to acquire a velocity in falling which would carry them uniformly through 32-2 feet in the next second, and an increase of velocity, equal to this, is found to be added to every succeeding second of time.

PROP. XXIV. The force of the attraction of gravitation acting upon any body is as its quantity of matter.

For each particle of matter in any body being acted upon by gravitation, the greater number of particles are contained in any body, the greater force must be exerted upon it; that is, the force increases as the quantity of matter increases.

EXP. Let two unequal balls, suspended by threads of the same length, be let fall at the same time from points equally distant from the lowest points of the arcs in which they move: The vibrations of each will be performed in equal times, and consequently their velocities will be equal; whence the momenta (Prop. XI.) will be as the quantities of matter; But (by Prop. XIII.) the force producing motion, is as the quantity of motion, or momentum, produced: Therefore the force of gravitation is as the quantity of matter; that is, as much greater force is exerted upon the larger body than upon the less, as its quantity of matter is greater than that of the less.

COR. I. The weight of any body is as its quantity of matter; for weight is the degree of force with which any body is acted upon by gravitation.

COR. 2. If the attraction of gravitation were increased in any ratio, the weight of a given body would be increased in the same ratio. Substituting, therefore, W, Q, F, for the weight, quantity of matter, and force of gravity, respectively, and supposing them to be variable; W will be as Qx F.

PROP. XXV. The velocities of bodies falling from the same height, without resistance, are equal.

If two bodies of different quantities of matter fall from the same height, the attracting force which acts upon the greater body, will (Prop. XXIV.) exceed that which acts upon the less, as much as the greater body exceeds the less in quantity of matter; whence they must move with equal velocities.

EXP. A guinea, and a feather, or other light body, in the exhausted receiver of an airpump, will fall through the same space in the same time.

PROP. XXVI. The spaces described by falling bodies are as the squares of the times from the beginning of the fall, and also as the squares of the last acquired velocities; or in the ratio compounded of the times and velocities.

In