Question 122: Motion in 1D

Question

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 __________

Accepted Answer

x
2
= at
2
+ 2bt + c .....(1)
$$ \Rightarrow $$ 2xv = 2at + 2b .....(2)
$$ \Rightarrow $$ v = $${{at + b} \over x}$$
differentiating (2) w.r.t. time
$$ \Rightarrow $$ xa' + v
2
= a
Here a' is acceleration.
$$ \Rightarrow $$ a'x = a - v
2
= a - $${\left[ {{{at + b} \over x}} ight]^2}$$
$$ \Rightarrow $$ a'x = $${{a{x^2} - {{\left( {at + b} ight)}^2}} \over {{x^2}}}$$
$$ \Rightarrow $$ a' = $${{a\left( {a{t^2} + 2bt + c} ight) - {{\left( {at + b} ight)}^2}} \over {{x^2}}}$$
$$ \Rightarrow $$ a' = $${{ac - {b^2}} \over {{x^3}}}$$
$$ 	herefore $$ a' $$ \propto $$ $${1 \over {{x^3}}}$$ $$ \propto $$ x
-3
$$ 	herefore $$ n = 3

Motion in 1D

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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