TAX

The body being projected in the direction of the line A T with a velocity = a, its velocity in the horizontal direction will continue uniform, and =a cos. A. In lilie manner, its velocity in the vertical direction A Y, due to the projectile force, is = a sin. A; and at the end of any time t the spaces pa ed over in thope directions, if gravity did not act, would be respectively, t a cos. z, and t a sin. A. But the space through which a heavy body falls by the action of gravity in the time < is ^g t^\ and as this is in the vertical direction, and opposite to A Y, it must be joined to the resolved part of the projectile motion in that direction, with a contrary sign. We have, therefore, X = t a cos. A, y—ta sin. k — \gt'^.... (1.) On eliminating t from these two equations, and supposing the velocity a to be that which a body would acquire in falling from a height = A, so that a=\/ 2g Ji, we obtain the following for the equation ©f the curve described by the projectile: — y=.tan.A_-^-^^....(2.) This equation belongs to a parabola whose axis is vertical^ or parallel to A Y. The summit of the parabola is found by differentiating the equation, and making — ^= 0, dx which gives x=2h cos. A sin. A, and consequently y = h sin."^ A, for the values ofx and y at that point. In order to find the amplitude or range of the projectile, that is, the point B in which it again pa es through the horizontal plane from which it was projected, we have only to suppose, in the above equation, y = 0. This gives X = 4 h [s. 1001]