# with the help of a graph, derive the relation v = u + at.

Consider the velocity-time graph of an object that moves under uniform acceleration as shown in the figure (u โ  O).

From this graph, we can see that initial velocity of the object (at point A) is u and then it increases to v (at point B) in time t. The velocity changes at a uniform rate a. As shown in the figure, the lines BC and BE are drawn from point B on the time and the velocity axes respectively, so that the initial velocity is represented by OA, the final velocity is represented by BC and the time interval t is represented by OC. BD = BC – CD, represents the change in velocity in time interval t. If we draw AD parallel to OC, we observe that

BC = BD + DC = BD + OA

Substituting, BC with v and OA with u, we get

v = BD + u

or BD = v u

Thus, from the given velocity-time graph, the acceleration of the object is given by

A = Changeย  inย  velocity \over Timeย  taken

= BD \over ADย  ย = BD \over OC

Substituting, OC with t, we get

a = BD \over ๐กย  or BD = at

From equations (1) and (2), we have

v โ u = atย  ย orย ย  v = u + at

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