When the collision takes place, both the falling body and the wedge will be compressed in the direction of their length. Let the linear contraction of P be represented by p, and that of the wedge, supposed at present immoveable, by q (both p and q coinciding in direction with Mee Then if * represent the modulus of elasticity (or the height of a vertical column of iron, having a base equal to the head of the wedge, whose weight would, by the compression it causes, reduce the height P or q of the supposed masses of iron to zero) and the forces of corn pression be assumed proportional to the contractions which they produce, we shall have and for the forces by which the falling body and the wedge are respectively compressed in consequence of the colliedon ; or the forces by which they tend to recover their original state : let these be represented by mp and nq respectively ; or, in terms of the force of gravity, by mgp and ngq. Then agq will represent the motive force by which the movement of the falling body is resisted after the impact, or — will represent the retardative force that body.
But from the equality of action and re-action we have mpeeng ; p , and p+q, or the aum of the compressions, is equal + to /X — g let this be represented by ; then q= — and re + /X vigs N (n+ ow, by dynamics, accelerative or retardative force Is des represented by v , r here being the velocity of the falling body at any time t between the instant of impact and that at which its motion ie extinguished by the resistance : therefore rile Vittf,11 (n +m) r' and integrating, v representing the velocity at the instant of impact at which tenet= 0, map (m + s) But when the wedge begins to move, the friction is equal to the force by which the falling body is compressed ; therefore, making Q /7111.1 (is en)
equal (=me) we have ; which being substituted in the above equation, we have (n+ m) gee ve=ve ma P Now the wedge being uniformly resisted by friction, while moving in consequence of the impact, the reterdative forco f, expressed in fe4 times of gravity, will ho w representing the weight of the wedge in terms homologous to r. Therefore since, by dynamics, a -- • if we represent the space through which the wedge moves in the direction me by z, we on substituting for e and f their values, and for putting its equivalent 22h, where It is the height duo to the velocity v, / z=(1. - 2meg r ) The values of m and re that is, of and , may be found, since e, the modulus for iron, is known to be about 10,000,000 feet ; an I consequently the relation between z and q can be determined In numbers.