Page:Elementary Text-book of Physics (Anthony, 1897).djvu/74

 particle along the $$x$$-axis is $$vt \cos{\alpha}.$$ If the force of gravity did not act on the particle its displacement along the $$y$$-axis in the same time would be $$vt \sin{\alpha};$$ but, since gravity acts, its real displacement along that axis is less than this by $$s = \tfrac{1}{2}gt^2,$$ where $$g$$ is the measure of the force or the acceleration of gravity, so that its displacement along the $$y$$-axis is $$vt \sin{\alpha} - \tfrac{1}{2}gt^2.$$ The path of the particle, or the series of points which it occupies at successive instants, is found by eliminating $$t$$ between the two equations for the two rectangular displacements. The equation of the path thus obtained is This represents a parabola passing through the origin. The axis is vertical, and the latus rectum is $$\frac{2v^2 \cos^2{\alpha}}{g}\cdot$$ If $$\alpha = 0,$$ or if the projection is horizontal, the equation becomes $$x^2 = - \frac{2v^2}{g}y,$$ representing a parabola with its vertex at the origin.

When the body is projected above the horizontal plane, so that $$\alpha$$ lies between zero and $$\frac{\pi}{2},$$ it will attain its greatest height at the instant when its velocity along the $$y$$-axis becomes zero, or when $$v \sin{\alpha} = gt.$$ The time required for it to describe its whole path and return to the $$x$$-axis is double this time or $$\frac{2v \sin{\alpha}}{g}\cdot$$. Its range, or the distance between its starting-point and the point at which it again meets the $$x$$-axis, is given by the product of this time and its horizontal velocity $$v \cos{\alpha},$$ or is $$\frac{v^2}{g}\sin{\alpha}\cos{\alpha} = \frac{v^2}{g} \sin {2 \alpha}.$$ The range is therefore a maximum when $$\alpha = 45^\circ$$. Since $$\sin{(\pi - 2 \alpha)} = \sin{2 \alpha},$$ the range is the same for projections at the angles $$\alpha$$ and $$90^\circ - \alpha,$$ or for projections equally inclined to the line bisecting the angle between the axes and on opposite sides of it.

'''53. Difference of Potential. The Potential.'''—Forces may arise from various causes. In any case they are only exhibited when