Prove first equation of motion​

Let us assume a body has initial velocity ' u ' and it is subjected to a uniform acceleration ' a ' so that the final velocity be ' v ' after a time interval ' t '. Acceleration is the rate of change of Velocity.Now , by the definition of acceleration , we have :

And we're done! ☂

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You need to be more specific.  i really don't know what a "motion map" is.you can draw a graph of an object's position, distance, displacement, speed,velocity, or acceleration versus time, or you can draw a graph of the object's x-position versus its y-position and not give any information about time at all.if the graph presents position or displacement against time,  then constant velocity is a straight line . . i.e. a line that has  constant slope.if the graph shows speed against time, then constant velocityis a horizontal piece of line.if the graph shows acceleration against time, then constant  velocity is a piece of line that lies on the x-axis . . it's value is zero.