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obtain equations of motion for constant acceleration using calculus method
Asked by klvnsnthl | 25 Dec, 2023, 07:50: PM

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)

------------------------------

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)

------------------------------------------------

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 .

Answered by Thiyagarajan K | 26 Dec, 2023, 12:01: AM

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