than A; required the proportion of the velocities with which they move?

Answ. velocity of A: velocity of B:: 3:40. (for v

vo

Er. 11. There are two bodies A and B, A contains 8 times the quantity of matter that B contains, and is moved with a force 48 times greater; the ratio of their velocities is required? Answ. velocity of A: that of B; : 6:1.

Ex. 12. The body A moves 40 times swifter than B, but A has moved only one minute, whereas B has been in motion 2 hours: the ratio of the spaces described by these bodies is required?

Answ. B has moved over 8 times more space than A.

Er. 13. The body A has passed over 50 miles, the body B only 5, but A moves with 5 times the velocity of B; what is the ratio of the times they have been in motion?

Answ. 2 to 1.

Ex. 14. The body A moves 30 times swifter than B and A has moved 12 minutes but B only 1, what difference will there be between the spaces described by them, supposing that B has moved over a space of 5 feet?

Answ. 1795 feet.

Ex. 15. The battering-ram of Vespasian weighed, suppose 100000lbs. and was moved, let us admit, with such a velocity by strength of hands as to pass through 20 feet in one second of time; and this was found to be sufficient to demolish the walls of Jerusalem. The question is, with what velocity a 32lbs, ball must move to do the same execution?

Answ. 62500 feet per second.

267. Scholium. It appears from art. 249, that when different forces act upon equal bodies, the forces are as the velocities communicated; therefore, since the velocity is proportional to the force, one may be substituted for the other, and all that has been said in the first book concerning the composition and resolution of forces, may be applied to the composition and resolution of velocities, or to the composition and resolution of motion.

It is but proper however, to remark, that the composition and resolution of forces, and the similar composition and resolution of motions, are distinct objects of enquiry; as the former. is a question entirely physical, the latter a problem purely mathematical. Some authors have inferred from their demonstrations of the latter problem the truth of the former, but this is certainly inconsistent with scientific precision, as each subject ought to be established on its own proper basis.

SECTION II,

On motions uniformly varied.

268.

PROPOSITION LXV.

The velocities generated in equal bodies, by the action of constant and uniform forces are as the forces and times the bodies are in motion.

For, when the times are equal, the velocities generated are as the forces which produce them; and when the forces are equal, the velocities are as the times the bodies have been in motion; therefore, the times and the forces being both of them different, the velocities will be as the forces and times jointly.

Thus, if V and v, be the velocities, Fand ƒ the forces, and T V FT x-x

t the times, then

to unity, then VxFxT.

and if v, f and t, are all equal

V

269. Cor. 1, Because VxFxT, we have T that is

TF

the time is as the velocity directly, and force inversely, And

V

in the same manner For the force is as the velocity

directly and time inversely. When F is constant V T.

270. Cor. 2. When a body in motion is retarded by the action of a uniform force, the velocity destroyed, in any given time, is as the force and time jointly.

271. Cor. 3. The momenta generated in unequal bodies, are as the forces and their times of acting jointly: i. c. Ma FT.

Consequently the momenta destroyed in any times are also as the retarding forces and their times of action.

PROPOSITION LXVI.

272. The spaces which bodies describe from quiescence, by constant and uniform forces, are as the times they have been in motion, and the last acquired velocities jointly.

For, in equal times, the spaces described will be greater or less according as the velocities of the moving bodies are greater or less; that is, the spaces will be as the velocities; also when the velocities are equal, the spaces evidently increase with the times the bodies are in motion; that is, the spaces are as the times.

Consequently, when neither the velocities nor the times are equal, the spaces are in a compound ratio of both: i. e. SxTV.

274. When the forces and bodies are the same, the spaces which bodies describe from rest, by the action of constant and uniform forces, are as the squares of the times they are in

motion.

For, by art, 272, the spaces described, estimated from quiescence, are as the last acquired velocities and the times the bodies have been in motion; and when the force is constant, by art. 269, the velocities are as the times: therefore, the spaces are as the squares of the times.

OTHERWISE; since by art. 272, SxTV, and by art. 268, V xFT, or V xT because F is given; therefore, if for V we put T in the expression SxTV, we shall have SxTT; that is, the spaces are as the squares of the times.

275. Cor. 1, When the forces are not equal then SF T2

276. Cor. 2. Because, by art. 269, when the force is

T V

given the velocity is proportional to the time, or x-;

t v

; that is, the spaces are also

****, we have

f

is, the accelerating forces are as the spaces directly, and as the squares of the times reciprocally; and when the times àre

F S

equal, the forces are as the spaces; that is also when

the spaces are equal, the forces are inversely as the squares of F t2

the times, or

278. Cor. 4. When bodies in motion are retarded by the actions of uniform forces, and move till their whole velocities are destroyed, the spaces described vary as the forces SFT X

and squares of the times; that is,

For the time in which any velocity is destroyed, is equal to the time in which the same velocity would be generated by the same force acting in a contrary direction.

V FT

TƒV

279. Cor. 5, By art. 268, X-X therefore, t

S ƒ V1 hence, (art. 275.) Thus, when bodies are retarded 7.

by uniform forces, and move till their whole velocities are destroyed, the spaces described are as the squares of the velocities directly, and as the retarding forces inversely.

Therefore, when bodies impinge against banks of earth, logs of wood, &c. where the retarding forces are uniform, the depths to which they sink, or the spaces described, are as the squares of the initial velocities directly, and the retarding forces inversely.