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Comparing with general quadratic equation $A{x}^{2}+Bx+C=0$ , we see that determinant of the corresponding quadratic equation is given by :
$$D={B}^{2}-4AC={u}^{2}-4X\frac{a}{2}X{x}_{0}$$ $$\Rightarrow D={u}^{2}-2a{x}_{0}$$
The important aspect of discriminant is that it comprises of three variable parameters. However, simplifying aspect of the discriminant is that all parameters are rendered constant by the “initial” setting of motion. Initial position, initial velocity and acceleration are all set up by the initial conditions of motion.
The points on the graph intersecting t-axis gives the time instants when particle is at the origin i.e. x=0. The curve of the graph intersects t-axis when corresponding quadratic equation (quadratic expression equated to zero) has real roots. For this, discriminant of the corresponding quadratic equation is non-negative (either zero or positive). It means :
$$D={u}^{2}-2a{x}_{0}\ge 0$$ $$\Rightarrow 2a{x}_{0}\le {u}^{2}$$
Note that squared term ${u}^{2}$ is always positive irrespective of sign of initial velocity. Thus, this condition is always fulfilled if the signs of acceleration and initial position are opposite. However, if two parameters have same sign, then above inequality should be satisfied for the curve to intersect t-axis. In earlier graphs, we have seen that parabola intersects t-axis at two points corresponding two real roots of corresponding quadratic equation. However, if discriminant is negative, then parabola does not intersect t-axis. Such possibilities are shown in the figure :
Clearly, motion of particle is limited by the minimum or maximum positions. It is given by :
$$x=-\frac{D}{4A}=-\frac{{u}^{2}-2a{x}_{0}}{4X\frac{a}{2}}=-\frac{{u}^{2}-2a{x}_{0}}{2a}$$
It may be noted that motion of particle may be restricted to some other reference points as well. Depending on combination of initial velocity and acceleration, a particle may not reach a particular point.
Our consideration, here, considers only positive values of time. It means that we are discussing motion since the start of observation at t=0 and subsequently as the time passes by. Mathematically, the time parameter, “t” is a non-negative number. It can be either zero or positive, but not negative. In the following paragraphs, we describe critical segments or points of a typical position – time graph as shown in the figure above :
Point A : This is initial position at t = 0. The position corresponding to this time is denoted as ${x}_{0}$ . This point is the beginning of graph, which lies on x-axis. The tangent to the curve at this point is the direction of initial velocity, “u”. Note that initial position and origin of reference may be different. Further, this point is revisited by the particle as it reaches E. Thus, A and E denotes the same start position - though represented separately on the graph.
Curve AC : The tangent to the curve part AB has negative slope. Velocity is directed in negative x-direction. The slope to the curve keeps decreasing in magnitude as we move from A to C. It means that speed of the particle keeps decreasing in this segment. In other words, particle is decelerated during motion in this segment.
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