Q. 13.6( 9 Votes )

Derive the three

Answer :

Equations of motion from velocity – time graph:


Graph shows the change in velocity with time for an uniformly accelerated object. The object starts from the point D in the graph with velocity u. Its velocity keeps increasing and after time t it reaches the point B on the graph.


The initial velocity of the object = u = OD = EA


The final velocity of the object = v = OC = EB


Time = t = OE = DA


Also from the graph we know that, AB = DC


First equation of motion


By definition, acceleration = change in velocity / time


= (final velocity – initial velocity)/ time


= (OC – OD) / OE


= DC / OE


a = DC / t


DC = AB = at


From the graph EB = EA + AB


v = u + at


This is first equation of motion


Second equation of motion


From the graph the distance covered by the object during time t is given by the area of quadrangle DOEB


s = area of the quadrangle DOEB


= area of the rectangle DOEA + area of the triangle DAB


= (AE × OE) + (1/2 × AB × DA)


s =


This is second equation of motion.


Third equation of motion


From the graph the distance covered by the object during time t is given by the area of the quadrangle DOEB.


Here DOEB is a trapezium.


Then,


S = area of trapezium DOEB


= 1/2 × sum of length of parallel side × distance between parallel sides


= 1/2 × (OD + BE) × OE


=


since


Therefore



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