An electron, starting from rest, has an acceleration that increases linearly with time; that is, a = kt, in which k = 1.50 m/s3 . (a) Plot a versus t during the first 10-s interval. (b)From the curve of part (a) plot the corresponding vversus t curve and estimate the electron’s velocity 5 s after the motion starts. (c) From the v versus t curve of part (b) plot the corresponding x versus t curve and estimate how far the electron moves dring the first 5 s of its motion.

(b) Graphically, the acceleration is given as the slope of the velocity–time graph. The slope of the velocity–time graph corresponds to the angle made by the tangent at any point on the curve with the x axis.Since the acceleration increases linearly with time, the slope of the velocity time curve must do the same. This is possible only when the velocity is dependent non-linearly on time, such that the velocity is proportional to the second power of variable time t.
The figure below shows three lines 1, 2, and 3 as tangents to different points on a parabolic curve, it can be seen that the angle made by tangent 3 is largest, while the angle made by tangent 1 is zero as it runs parallel to the time axis.




Therefore, one can see that the slope of the parabolic curve increases with time and hence the velocity must be proportional to the second power of time t .
Therefore, the velocity is given as


(c) Graphically, the velocity is given as the slope of the position–time graph. The slope of the position time graph corresponds to the angle made by the tangent at any point on the curve with the x axis.Since the velocity increases non-linearly with time, the slope of the velocity time curve must do the same. This is possible only when the position is dependent non-linearly on time, such that the position is proportional to the third power of variable time t thereby accounting for the fact that velocity depends on the second power of time; that is,