Question

derive the formula: S=ut+1/2atsq.,where the symbols have usual meanings

Answer 1

Terminology : a = acceleration, v = final velocity , u = intial velocity, t = time.

From the definition of Acceleration ,

a = v - u / t.

Now, Transposing t to Left hand side.

at = v - u

Transpose -u to Left hand side.

at + u = v.

Interchanging the sides of the equation,

v = u + at.

Now, We have two velocities " u " & " v " .

Average velocity = u + v /2 .

We know that,
Velocity = Distance / Time

Distance = Velocity * Time

s = ( u + v) t / 2

Substitute v = u + at.

s = ( u + u + at ) t /2

s = ( 2u + at) t /2

s = ut + 1/2 at² .

Hence proved !

Answer 2

Hi.

Here is your answer---

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To Prove,
      S = ut + (1/2)at²
Where, S = Diatance
            u = Initial Velocity
            a = acceleration
             t = time.

Proof : 
 Acceleration = (Final velocity - Initial Velocity)/t
            a = (v - u)/t
             at = v - u -------------------------------eq(i)

Now, We know, 
  Distance Traveled = Average Velocity × Time
                           S  = (v + u)/2 × t
                               
But From eq(i) , v = u + at
                          S = (u + at + u)/2 × t
                          S = [(2u + at)/2] × t
                     ⇒  S = ut + (1/2)at²
                                                          Hence Proved.


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Hope it helps.

Have a nice day.