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Trajectory as seen by the person on ground
Without overemphasizing, we must acknowledge that the concept of path or trajectory is essentially specific to the frame of reference or the coordinate system attached to it. Interestingly, we must be aware that this particular observation happens to be the starting point for the development of special theory of relativity by Einstein (see his original transcript on the subject of relativity).
The position in three dimensional motion involves specification in terms of three coordinates. This requirement poses a serious problem, when we want to investigate positions of the object with respect to time. In order to draw such a graph, we would need three axes for describing position and one axis for plotting time. This means that a position – time plot for a three dimensional motion would need four (4) axes !
A two dimensional position – time plot, however, is a possibility, but its drawing is highly complicated for representation on a two dimensional paper or screen. A simple example consisting of a linear motion in the x-y plane is plotted against time on z – axis (See Figure).
Two dimensional position – time plot
As a matter of fact, it is only the one dimensional motion, whose position – time plot can be plotted conveniently on a plane. In one dimensional motion, the point object can either be to the left or to the right of the origin in the direction of reference line. Thus, drawing position against time is a straight forward exercise as it involves plotting positions with appropriate sign.
Problem : A ball moves along a straight line from O to A to B to C to O along x-axis as shown in the figure. The ball covers each of the distance of 5 m in one second. Plot the position – time graph.
Motion along a straight line
Solution : The coordinates of the ball are 0,5,10, -5 and 0 at points O, A, B, C and O (on return) respectively. The position – time plot of the motion is as given below :
Position – time plot in one dimension
A ball falling from a height ‘h’ strikes a hard horizontal surface with increasing speed. On each rebound, the height reached by the ball is half of the height it fell from. Draw position – time plot for the motion covering two consecutive strikes, emphasizing the nature of curve (ignore actual calculation).
Now, as the ball falls towards the surface, it covers path at a quicker pace. As such, the position changes more rapidly as the ball approaches the surface. The curve (i.e. plot) is, therefore, steeper towards the surface. On the return upward journey, the ball covers lesser distance as it reaches the maximum height. Hence, the position – time curve (i.e. plot) is flatter towards the point of maximum height.
Position – time plot in one dimension
The figure below shows three position – time plots of a motion of a particle along x-axis. Giving reasons, identify the valid plot(s) among them. For the valid plot(s), determine following :
Position – time plots in one dimension
Validity of plots : In the portion of plot I, we can draw a vertical line that intersects the curve at three points. It means that the particle is present at three positions simultaneously, which is not possible. Plot II is also not valid for the same reason. Besides, it consists of a vertical portion, which would mean presence of the particle at infinite numbers of positions at the same instant. Plot III, on the other hand, is free from these anomalies and is the only valid curve representing motion of a particle along x – axis (See Figure).
Valid position – time plot in one dimension
When the particle comes to rest, there is no change in the value of “x”. This, in turn, means that tangent to the curve at points of rest is parallel to x-axis. By inspection, we find that tangent to the curve is parallel to x-axis at four points (B,C,D and E) on the curve shown in the figure below. Hence, the particle comes to rest four times during the motion.
Positions of rest
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