A particle of mass \(0.1 \mathrm{~kg}\) is subjected to a force which varies with distance as shown in figure. If it starts its journey from rest at \(\mathrm{x}=0\). What is the particle's velocity square at $\mathrm{x}=6 \mathrm{~cm}$ ? (A) \(0(\mathrm{~m} / \mathrm{s})^{2}\) (B) \(240 \sqrt{2}(\mathrm{~m} / \mathrm{s})^{2}\) (C) \(240 \sqrt{3}(\mathrm{~m} / \mathrm{s})^{2}\) (D) \(480(\mathrm{~m} / \mathrm{s})^{2}\)

Convert the given distance x = 6 cm into meters (SI unit for distance) by dividing it by 100: \[x = \frac{6}{100} \mathrm{m} = 0.06 \mathrm{m}\]

We can find the area under the force-displacement curve by dividing the entire area into triangles and rectangles. The figure shows a right triangle with its base along the x-axis and its height along the y-axis. The base has a length of 0.06 m (from step 1) and the height is given by the force at x=0.06 m, that is, \(F = -40 \mathrm{~N}\). The negative sign indicates that the force opposes the displacement. The area of a triangle is given by \(\frac{1}{2} \times base \times height\), thus the area under the curve (which represents work done) is \[W = -\frac{1}{2} \times 0.06 \mathrm{m} \times 40\mathrm{~N} = -1.2\mathrm{~J}\]

According to the work-energy theorem, the work done on the particle equals the change in its kinetic energy. The particle starts its journey from rest, so the initial kinetic energy is \(0 \mathrm{~J}\). Let the final kinetic energy be \(KE_f\), then \[W = KE_f - KE_i\] Since the initial kinetic energy is 0, we have \[W = KE_f\] Substitute the work done value found in step 2: \[-1.2\mathrm{~J} = KE_f\]

The kinetic energy of the particle is given by \(\frac{1}{2}mv^2\), where m is the particle's mass and v is its velocity. Now we have \(KE_f = -1.2 \mathrm{~J}\), and the particle's mass is \(0.1 \mathrm{~kg}\). Solving for the particle's velocity square, we get \[\frac{1}{2}(0.1 \mathrm{~kg})v^2 = -1.2 \mathrm{~J}\] \[v^2 = \frac{-2 \times 1.2 \mathrm{~J}}{0.1 \mathrm{~kg}}\] \[v^2 = -24 (\mathrm{~m} / \mathrm{s})^{2}\] The square of the particle's velocity at x=6 cm is \(-24 (\mathrm{~m} / \mathrm{s})^{2}\). Since this value is negative, it indicates that there's an inconsistency in the given values or the figure. It could be a mistake in the problem statement or a misinterpretation of the figure. This result doesn't match any of the given options (A, B, C, or D).