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Derive the equation of motion of a body in uniform acceleration using graphical method
Asked by mateilrf | 19 Jan, 2021, 10:25: PM
Acceleration a is defined as rate of change of velocity.

If at initial time to , let the velocity be u and after time t , let the velocity be v.

Acceleration a = ( v - u ) / ( t - to )   ..................(1)

If acceleration is uniform then a is constant with respect to time.

From eqn.(1) , we get v =  u + a ( t - to ) .................(2)

Let us consider we start recording time from reference time instant to , so that we consider to = 0

Hence we get velocity as a function of time as , v = u + a t  ...................(3)

Graph of eqn.(3) that gives velocity as a function of time is linear and it is given below

Area under the velocity-time graph between time τ =0 to time τ = t is the distance travelled by the object in time duration of t seconds.

As seen from graph given above, area under the graph between time τ = 0 to time τ = t is a trapezium OABC

Area of trapezium OABC = (1/2) [ OA + BC ] × OC  ......................(4)

OA = inital velocity  u  ;  BC = final velocity v  ; OC = time duration t

Hence eqn.(4) becomes ,   Area OABC = (1/2) [ u+v ] × t

If S is distance travelled by object in time duration t , then above equation becomes ,

S = (1/2) [ u+v ] × t ........................(5)

If we substitute final velocity v from eqn.(3) , we get , S = (1/2) [ u + u + a t ] × t

Hence we get , S = u + (1/2) a t2 .........................(6)

If we want to eliminate t from eqn.(5) , we use eqn.(3) to substitute t as , t = ( v-u) /a ............(7)

Using eqn.(7) , we rewrite eqn.(5) as

S = (1/2) [ u + v ] ( v - u ) /a   or     v2 = u2 + ( 2 a S )   ............................(8)

Eqn.(3) , Eqn.(6) and Eqn.(8) are equations of motion if acceleration is uniform

v = u + a t  .................................(3)

S = u + (1/2) a t2 .........................(6)

v2 = u2 + ( 2 a S )   ......................(8)
Answered by Thiyagarajan K | 20 Jan, 2021, 10:36: AM

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