Question

Prove S = ut + 1/2 at2 where S, u, t, and a have their usual meaning​

Answer 1

Answer:

Suppose a body has an initial velocity 'u' and a uniform acceleration 'a' for time 't' so that its final velocity becomes 'v'. Let the distance travelled by the body in this time be 's'. The distance travelled by a moving body in time 't' can be found out by considering its average velocity. Since the initial velocity of the body is 'u' and its final velocity is 'v', the average velocity is given by  

Average velocity =  Initial velocity + Final velocity/2

​ That is, Average velocity =   u+v/2

​ Also, Distance travelled = Average velocity × Time  

So, s=(   u+v/2 )×t

From the first equation of motion, we have, v=u+at.

Put this value of v in equation (1), we get:  

s=(   u+u+at/2 )×t

or s=  (2u+at)×t/2

​ or  s=   2ut+at2/2  

or s=ut+ 1/2 at2  

​ where, s= distance travelled by the body in time t

u= initial velocity of the body  

and a= acceleration

Answer 2

The initial velocity u at point A changes at a uniform rate from A to B in time t. BC is the final velocity v in the graph. The time t is represented by OC. The distance travelled by the body is the area under the OABC.

Therefore, distance travelled = area of under OABC

= area of rectangle OADC + area of triangle ABD

Area of rectangle OADC and the area of triangle ABD is given as:

Area of rectangle OADC = (OA)(OC) = ut

Area of triangle ABD = (½)(area of rectangle AEBD) = ½ at²

Therefore, distance travelled, s = ut + ½ at²

here s= distance

u= initial velocity

t= time

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