Page:Elementary Text-book of Physics (Anthony, 1897).djvu/31

point $A$. Draw the lines $PQ$ and $PR$ in the directions of the tangents at $A$ and $B$, equal to the velocities $v_{0}$ and $v$ of the point at $A$ and $B$ respectively. The line QB is the change in the velocity of the point during the time in which it traverses the distance $AB$. Draw the line $QS$ perpendicular to $PR$. The angle $QPR$, being the angle between the tangents at $A$ and $B$, equals the angle $\alpha$. In the limit, as $\alpha$ vanishes, $v$ and $v_{0}$ differ by the infinitesimal $SR$, and $QS$ equals $v\alpha$. The line $SB$ represents the change in the numerical magnitude of the velocity during the time $t-t_{0}$, and the rate of that change, which takes place along the tangent to the path, is given by

$a_{t}={\frac {v-v_{0}}{t-t_{0}}}\cdot .$

(5)

The line $QS$ represents the change in velocity during the same time along the normal to the path. The acceleration along that normal is therefore ${\frac {v\alpha }{t-t_{0}}}\cdot$. Now under the conditions assumed in these statements $AB=r\alpha$, and ${\frac {AB}{t-t_{0}}}=v$, the velocity of the point. Hence $v={\frac {r\alpha }{t-t_{0}}}$, and the acceleration along the normal to the path is

$a_{n}={\frac {v^{2}}{r}}\cdot$

(6)

If the path be a straight line, the normal acceleration vanishes, and the whole acceleration is given by the limit of the ratio ${\frac {v-v_{0}}{t-t_{0}}}={\frac {dv}{dt}}\cdot$ If the path be a circle, and if the point move in it uniformly, the whole acceleration is given by ${\frac {v^{2}}{r}}$.

The unit of acceleration is that of a point, the velocity of which changes at a uniform rate by one unit of velocity in one second.

The dimensions of acceleration are $LT^{-2}$. Acceleration is completely described when its magnitude and