Question

Draw a velocity time graph of a uniformly accelerated motion and derive equation of motion,
$s = ut + \dfrac{1}{2}a{t^2}$from it.

Answer 1

Velocity-time graph of an object moving with uniform acceleration. When the velocity – time graph is plotted for an object moving with uniform acceleration, the slope of the graph is a straight line, the pattern of slope of the graph shows that object is moving with uniform acceleration.

Complete step by step answer:

In the sequence,
$S$ is displacement,
$u$ is initial velocity,
$t$ is time,
$a$ is acceleration
$V$is final velocity
${V_{av}}$angle velocity
From that,
$av$ velocity ${v_{av}} = \dfrac{{u + v}}{2}$. . . (1)
${V_{av}} = \dfrac{S}{t}$. . . (2)
From (1)
$\dfrac{s}{t} = \dfrac{{u + v}}{2}$ multiply
$\to S = \dfrac{{\left( {u + v} \right)t}}{2}$. . . (3)
From (2) equation of motion
$v = u + at$. . . (4)
Resulting value of (4) and (3)
$S = \dfrac{{(u + u + at)t}}{2}$
$\Rightarrow S = \dfrac{{ut + ut + a{t^2}}}{2}$ by separating the equation,
$S = \dfrac{{2at}}{2} + \dfrac{{a{t^2}}}{2} \\\ \implies S = ut + \dfrac{1}{2}a{t^2} \\\$

Additional Information:
Advantage of speed time graph speed-time graphs are very useful when describing the movement of an object. This is a velocity time graph of an object moving in a straight line due to North. The displacement of this object is the area of the velocity time graph. A sloping line on a speed-time graph represents acceleration.

Note:
if an object's speed is increasing at a constant rate then we say it has uniform acceleration; the rate of acceleration is constant. If a car speeds up, then slows down then speeds up doesn’t have uniform acceleration. Non-uniform means that the acceleration is not uniform. Uniform acceleration is change of equal velocity in equal intervals of time. Non-uniform acceleration is change of non-equal velocity in equal intervals of time.