"The distance x covered by a particle in one dimensional motion varies with time t as x2 = at2 + 2bt + c. If the acceleration of the particle depends on x as x–n, where n is an integer, the value of n is __________"

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"x² = at² + 2bt + c .....(1)

$\Rightarrow$ 2xv = 2at + 2b .....(2)

$\Rightarrow$ v = ${{at + b} \over x}$

differentiating (2) w.r.t. time

$\Rightarrow$ xa' + v² = a

Here a' is acceleration.

$\Rightarrow$ a'x = a - v² = a - ${\left[ {{{at + b} \over x}} \right]^2}$

$\Rightarrow$ a'x = ${{a{x^2} - {{\left( {at + b} \right)}^2}} \over {{x^2}}}$

$\Rightarrow$ a' = $${{a\left( {a{t^2} + 2bt + c} \right) - {{\left( {at + b} \right)}^2}} \over {{x^2}}}$$

$\Rightarrow$ a' = ${{ac - {b^2}} \over {{x^3}}}$

$\therefore$ a' $\propto$ ${1 \over {{x^3}}}$ $\propto$ x^-3

$\therefore$ n = 3"

The distance x covered by a particle in one dimensional motion varies with time t as
x² = at² + 2bt + c. If the acceleration of the particle depends on x as x^–n, where n is an integer, the value of n is __________

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The distance x covered by a particle in one dimensional motion varies with time t as x2 = at2 + 2bt + c. If the acceleration of the particle depends on x as x–n, where n is an integer, the value of n is __________

Solution

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The distance x covered by a particle in one dimensional motion varies with time t as
x² = at² + 2bt + c. If the acceleration of the particle depends on x as x^–n, where n is an integer, the value of n is __________