Question

Derive displacement-time relation from velocity-time graph
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Answer 1

Let an object is moving with uniform acceleration.

Let the initial velocity of the object = u

Let the object is moving with uniform acceleration, a.

Let object reaches at point B after time, t and its final velocity becomes, v

Draw a line parallel to x-axis DA from point, D from where object starts moving.

Draw another line BA from point B parallel to y-axis which meets at E at y-axis.Let OE = time, t

Now, from the graph,

BE = AB + AE

⇒ v = DC + OD (Since, AB = DC and AE = OD)

⇒ v = DC + u (Since, OD = u)

⇒ v = DC + u ------------------- (i)

Now, Acceleration (a) =(Change in velocity)/(Time taken)

a= v – ut

⇒ a = v - ut

⇒ a= OC – OD t = DC t

⇒a = OC-OD t =DC t

⇒at=DC

⇒at=DC -----(ii)

By substituting the value of DC from (ii) in (i) we get

v= at + u

v = at + u

⇒ v = u + at

⇒ v = u + at

Above equation is the relation among initial velocity (u), final velocity (v), acceleration (a) and time (t). It is called first equation of motion.