Explanation:
Given : speed \({=v}\), stopping distance \({=x}\) kinetic energy of body, \(K = \frac{1}{2}m{v^2}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\,\,\,(1)\)
Now, kinetic energy is doubled, \(2K = \frac{1}{2}m{v'^2}\quad \quad \quad \quad (2)\)
From eq.(1) and (2),we get \({v^\prime } = \sqrt 2 v\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)\)
Now, from eq. of motion
\({v^2} - {u^2} = 2{\rm{ as }}\)
\( \Rightarrow 0 - {v^2} = - 2ax\) \(\quad \quad \) (final velocity \( = 0\))
\( \Rightarrow {v^2} = 2ax\quad \quad \quad \quad (4)\)
When kinetic energy is doubled, let new stopping distance be \(x'{\mkern 1mu} {(\sqrt 2 v)^2} = 2ax'\quad \quad \quad (5)\)
Dividing eq.(4) eq.(5), we get
\(\dfrac{1}{2}=\dfrac{x}{x^{\prime}} \Rightarrow x^{\prime}=2 x\).
So correct option is (1)