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Fig. 2 (NCT). 6 shows x-t plot of one d...

Fig. 2 (NCT). 6 shows ` x-t` plot of one dismensional motion a particle. Is it correct to say from the graph that the particle moves in a straight line for ` t lt 0 `and on a parabolic path form ` t gt 0 `? If not, suggest a suitable physical contxt for this graph.

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No, it is not correct to say that a particle moves in a straight line for `t lt 0` and on a parabolic path for `t gt 0`. Because the x-t graph cannot show the trajectory of the particle. In this graph when t=0, x=0, the situation can be correlated to the motion of a body dropped freely from a height.
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