Question

define the expression
$s = ut + \frac{1}{2} {at}^{2}$
in Mathematical and in graphical​

Answer 1

Answer:

Velocity is defined as the rate of change of displacement. This is mathematically represented as:

Velocity=DisplacementTime

Rearranging, we get

Displacement=Velcoity×Time

If the velocity is not constant then in the above equation we can use average velocity in the place of velocity and rewrite the equation as follows:

Displacement=(InitialVelocity+FinalVelocity2)×Time

Substituting the above equations with the notations used in the derivation of the first equation of motion, we get

s=u+v2×t

From the first equation of motion, we know that v = u + at. Putting this value of v in the above equation, we get

s=u+(u+at))2×t

s=2u+at2×t

s=(2u2+at2)×t

s=(u+12at)×t

On further simplification, the equation becomes:

s=ut+12at2

Answer 2

Answer:

Derivation of Second Equation of Motion by Graphical Method

From the graph above, we can say that

Distance travelled (s) = Area of figure OABC = Area of rectangle OADC + Area of triangle ABD

s=(½AB×BD)+(OA×OC)

Since BD = EA, the above equation becomes

s=(½AB×EA)+(u×t)

As EA = at, the equation becomes

s=½×at×t+ut

On further simplification, the equation becomes

s=ut+½at²