Mizoram Class 9 Answered
Derive the equation of motion of a body in uniform acceleration using graphical method
Asked by mateilrf | 19 Jan, 2021, 10:25: PM
Expert Answer
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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Mizoram 9 - Physics
Asked by mateilrf | 19 Jan, 2021, 10:25: PM
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