A text book of elementary mechanics, for the use of colleges and schools . e resistance of the air be neglected, thepath of a projectile is a parabola. Suppose a body start from A in the direction AD(Fig. 21) with an initial velocity u; at the end of tseconds, if no other motion were imparted to it, it wouldreach a point D, so that AD = ut. (1) 44 KINEMATICS. [48. But from the instant of starting it falls vertically-downward under the influence of gravity, with uni-formly accelerated motion. At the end of t seconds, ifit had no initial velocity, it would fall to B, so that AB = igt\ (2) But as
A text book of elementary mechanics, for the use of colleges and schools . e resistance of the air be neglected, thepath of a projectile is a parabola. Suppose a body start from A in the direction AD(Fig. 21) with an initial velocity u; at the end of tseconds, if no other motion were imparted to it, it wouldreach a point D, so that AD = ut. (1) 44 KINEMATICS. [48. But from the instant of starting it falls vertically-downward under the influence of gravity, with uni-formly accelerated motion. At the end of t seconds, ifit had no initial velocity, it would fall to B, so that AB = igt\ (2) But as shown above (46), as the body must obey bothtendencies to motion simultaneously, its actual the given time will be at C. Squaring (1) and dividing by (2), we have AD* _ W t£f _ 2^2 AB \ABr iff g or BO 2v? 9 . AB Therefore the ratio of the square of the ordinate BC(BC, BC) to the abscissa AB (AB, AB) is con-stant, and hence the curve is a parabola. 48. Position of the Directrix, Axis, Focus. The lineAD (Fig. 22) is a tangent to the parabola at A, the verti-. Fig. 22. cal line FAB is a diameter. The constant value of the is four times the distance from A ratio of BC2 2ri A B \ 9 to the directrix or to the focus. The same relation is at 49.] PEOJECTILES. 45 once obyious in the case of the parabola, whose equationis if = iax ( —- = ia); it may also be proved analyti-cally for this case, where the co-ordinates EB, AD arcoblique. It is proved geometrically in a following para-graph (50, e). If AE is taken on the vertical line equal to —, and EGE be drawn horizontally, this line will be the direc-trix; and if from A on the line AF (drawn so that the , u2angle EAD = DAF) we take AF = —, the point i^is the focus of the parabola. The vertical line GML isthe axis. If the direction of the initial velocity be horizontal, asin Fig. 23, then the starting-point ^A is the vertex, the vertical line AB *is the axis, .and the focus and direc- ?* 2? 2?
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectmechanics, bookyear18
