Question

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ii part​

Answer 1

To derive $\boxed {s \: = \: ut \: +\dfrac {1}{2}ut^2}$ by Graphical method

Suppose, the body travels the distance (s) and (t)

We know that :

Area of space between the velocity-time graph AB and the time axis OC which is equal to the area of the figure OADC

Thus,

Distance travelled = Area of figure OADC

= Area of rectangle OADC + Area of triangle ABD...(1st eq.)

Area of rectangle OADC = OA × AC

= u × t

= ut...(2nd eq.)

Area of triangle ABD = $\dfrac {1}{2}$ × Area of rectangle AEBD

= $\dfrac {1}{2}$ × AD × BD

= $\dfrac {1}{2}$ × t × at

= $\dfrac {1}{2}$ $at^2$

From eq. 1st,

s = Area of rectangle OADC + Area of triangle ABD

$\boxed {s \: = \: ut \: +\dfrac {1}{2}ut^2}$

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Answer 2

Consider an object is moving with a uniform acceleration “a” along a straight line. The initial and final velocities of the object at time t = 0 and t = t are u and v respectively. During time t, let s be the total distance travelled by the object. In uniformly acceleration motion the velocity – time graph of an object is a straight line, inclined to the time axis.

ATTACHMENT

Second Equation of Motion by Graphical Method

OD = u, OC = v and OE = DA = t.

Let, the Initial velocity of the object = u

Let, the object is moving with uniform acceleration, a

Let, the object reaches at point B after time t, and its final velocity becomes, v.

Draw a line parallel to x-axis DA from point, D from where object starts moving.

Draw another line BA from point B parallel to Y-axis which meets at E at y-axis.

Second Equation of Motion: Distance covered by the object in the given time “t” is given by the area of the trapezium ABDOE.

Let in the given time (t), the distance covered by the moving object = s

The area of trapezium, ABDOE.

Distance (s) = Area of ΔABD + Area of ADOE.

s = ½ x AB x AD + (OD x OE)

s = ½ x DC x AD + (u x t) [∵ AB = DC]

s = ½ x at x t + ut [∵ DC = at]

s = ½ x at x t + ut

s = ut + ½ at².

It is the expression gives the distance covered by the object moving with uniform acceleration.