Q. 383.9( 9 Votes )

Show by means of graphical method that :

v = u + at where the symbols have their usual meanings.

Answer :


The body has an initial velocity u at point A and then its velocity changes at a uniform rate from A to B in time t. In other words, there is a uniform acceleration 'a' from A to B, and after time t its final velocity becomes 'v' which is equal to BC in the graph. The time t is represented by OC. To complete the figure, we draw the perpendicular CB from point C, and draw AD parallel to OC. BE is the perpendicular from point B to OE.


Now, Initial velocity of the body, u = OA...... (1)


And, Final velocity of the body, v = BC........ (2)


But from the graph BC = BD + DC


Therefore, v = BD + DC ......... (3)


Again DC = OA


So, v = BD + OA


Now, From equation (1), OA = U


So, v = BD + u ........... (4)


We should find out the value of BD now. We know that the slope of a velocity – time graph is equal to acceleration, a.


Thus, Acceleration, a = slope of line AB


or a = BD/AD


But AD = OC = t,


so putting t in place of AD in the above relation, we get:


a = BD/t


or BD = at


Now, putting this value of BD in equation (4) we get :


v = at + u


This equation can be rearranged to give:


v = u + at


And this is the first equation of motion. It has been derived here by the graphical method.


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