Uniform Velocity & Acceleration

⁕ Uniform Velocity

↪ An object is said to be moving with a uniform velocity if it travels equal distances in equal intervals of time always moving in the same direction.
↪ This means that the displacement is directly proportional to the time and hence for a uniform velocity, a graph of the displacement and time is a straight line passing through the origin O as shown in figure.

↪ Let us draw QP and QR normal to X- and Y-axes respectively. 

↪ From right ∆OPQ, the slope of the straight line is

or, slope = velocity
and
displacement = velocity × time
or, s = ut

↪ This shows that the s-t graph is a straight line passing through the origin O and the slope gives the uniform velocity. 

↪ If an object moves with uniform velocity, the v-t graph is a straight line RP parallel to the time axis.
↪ Let us draw PQ normal to the time axis.
∴ the velocity, u = OR = QP and time, t = OQ
↪ Since it is known that velocity = displacement/time,
↪ then the displacement (s) is displacement = velocity × time or s = (QP)(OQ) = area of rectangle OQPR
↪ Thus the area under the v-t graph gives the displacement. 

⁕ Uniform Acceleration 

↪ If the velocity of an object is non-uniform, the object is said to be accelerated.
↪ An object is said to be moving with a uniform acceleration if its velocity changes by equal amounts in equal intervals of time.
↪ This means that the velocity is directly proportional to the time and hence for a uniform acceleration, a graph of the velocity and time is a straight line passing through the origin O as shown in figure.

↪ Let us draw QP and QR normal to X- and Y-axes respectively.

↪ From right ∆OPQ, the slope of the straight line is 

or slope = acceleration

↪ Thus the slope of the v-t graph gives the acceleration of the object with uniform acceleration.

↪ If the object moves with a uniform acceleration, the v-t graph will be a straight line with a positive slope.

↪ Its slope is equal to the acceleration.

↪ If the object moves with a uniform deceleration, the v-t graph will be a straight line with a negative slope.

↪ Its slope is equal to the negative acceleration or deceleration. 

↪ If the velocity of an object increases from its initial velocity u to final velocity v in time t,
then the displacement is
displacement = average velocity × time