A particle moves rectilinearly. It's displacement X at time t is given by x^2 = at^2 + b. It's acceleration at time t is proportional to:
Given the displacement as :-
x^2 = at^2 + b.
Differentiating it with respect to time we get :-
2xdx = 2atdt + 0
Rearranging the terms we have :-
dx/dt = at/x (dx/dt is velocity) .... 1
Now x from the given equation can be written as x = √(at^2 + b)
Substituting value of x from above to equation number 1 we get :-
dx/dt = at/(√at^2 + b) ...... 2
Now again differentiating equation 2 with respect to time (derivative of velocity is acceleration) we get :-
dv/dt = A( acceleration )
{ (√at^2 +b)*a - a^2*t^2/(√at^2 +b) }/ (at^2 + b)
Therefore , A = ab/[(at^2 + b)^3/2] .... 3
Now since x = √(at^2 + b) ....4
From equation 3 and 4 Acceleration is proportional to 1/(x^3).
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