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THEORY AND PRACTICE

TRACT 37.

PROBLEM XVIII.

158. To determine the Circumstances of Space, Penetration,Velocity, and Time , arising from a Ball moving with a givenVelocity, and striking a moveable Block of IVood, or otherSubstance.

*

B \

c

D

■ - X

\

I

Let the ball move in the direction ae, passing through thecentre of gravity of the block b, impinging on the point c;and when the block has moved through the space cd, inconsequence of the blow, let the ball have penetrated to thedepth de.

Let b == the mass or matter in the block,b — the same in the ball,

5 = CD the space moved by the block,

.<* — de the penetration of the ball, and theref.s -+■ x — ce the space described by the ball,a — the first velocity of the ball,v — the velocity of the ball at e,u = veloc. of the block at the same instant,t — the time of penetration, or of the motion,r = the resisting force of the wood.

Then shall — be the accelerating force of the block,

and — the retarding force of the ball.

Now, because the momentum me, communicated to theblock in the time t, is that which is lost by the ball, namely,_ bv, therefore 311 = - bi, and mi _ bv. But whenv = a, u = 0; therefore, by correcting, me — b (a — v)',or the momentum of the block is every where equal to themomentum lost by the ball. And when the ball has pene-trated to the utmost depth, or when u — v, this becomesb u = b (a - u), or ab = (b+£) u; that is, the momentum