CBSE Class 11-science Answered
Let v be the velocity that varies with time. Acceleration α is defined as rate of change of velocity .
Hence we have
By integrating above expresion we get
.........................(1)
where c is constant of integration which is to be determined from initial value of velocity
Let u be the velocity at time t = 0 . Using this initial condition in eqn.(1) , we get
u = 0 + c
Hence eqn.(1) becomes , ...................................(2)
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Let v be the velocity at a instant of time t .
Displacement dS made in small time interval is
By substituting velocity v from eqn.(2) , above expression is written as
By integrating above expression , we get
..........................................................(3)
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Displacement S made in time duration t is written as
As shown in figure, displacement S made between instants of time t'=0 and t'=t is area under the velocity function in velocity-time graph.
Let initial velocity be u and final velocity be v . When acceleration is uniform , velocity varies linearly with respect to time.
Hence area under the velocity function is trapezium . hence we write the displacement S as
S = (1/2) ( v + u) t ..............................(4)
Using eqn.(2) , time t is written as , t = ( v-u) /α ........................(5)
By substituting time t from eqn.(5) , we rewrite eqn.(4) as
S = (1/2) ( v + u ) [ (v-u) / α
Hence from above expression , we write v2 = u2 + ( 2 α S ) ...............................(6)
Eqn.(2) , (3) and (6) are the fundamental equations of uniformly accelerated motion .