### Related Formula v_1' = v_2 quad textand quad v_2' = v_1$v_1' = v_2 \quad \text{and} \quad v_2' = v_1$ for perfectly elastic collision (e=1$e=1$) when masses are identical (m_1 = m_2$m_1 = m_2$). ### Core Logic Reason (R) states that identical masses exchange their velocities during elastic collision, which is mathematically correct. Let us trace the sequence of collisions chronologically: 1. Since v_A = 5mathrm~m/s$v_A = 5\mathrm{~m/s}$ and v_B = 2mathrm~m/s$v_B = 2\mathrm{~m/s}$, sphere A collides with sphere B. After this collision, they swap velocities: v_A' = 2mathrm~m/s, quad v_B' = 5mathrm~m/s$v_A' = 2\mathrm{~m/s}, \quad v_B' = 5\mathrm{~m/s}$ 2. Now sphere B has velocity v_B' = 5mathrm~m/s$v_B' = 5\mathrm{~m/s}$ and sphere C has v_C = 4mathrm~m/s$v_C = 4\mathrm{~m/s}$. Sphere B will collide with C. After swapping: v_B'' = 4mathrm~m/s, quad v_C' = 5mathrm~m/s$v_B'' = 4\mathrm{~m/s}, \quad v_C' = 5\mathrm{~m/s}$ 3. Looking at the values now: v_A' = 2mathrm~m/s$v_A' = 2\mathrm{~m/s}$ and v_B'' = 4mathrm~m/s$v_B'' = 4\mathrm{~m/s}$. No more collisions occur. Therefore, the final velocities are v_A = 2mathrm~m/s, v_B = 4mathrm~m/s, v_C = 5mathrm~m/s$v_A = 2\mathrm{~m/s}, v_B = 4\mathrm{~m/s}, v_C = 5\mathrm{~m/s}$. The values given in Assertion (A) are wrong. Hence, (A) is false but (R) is true.

### Pattern Recognition Velocity exchange happens pairwise in sequential order. Do not try to solve simultaneous conservation laws across all three blocks at once; handle each collision step-by-step from left to right. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Work, Energy and Power

### Core Logic Let's apply the comprehensive Work-Energy theorem framework across key layout tracking nodes:

mathrmW_textall = Delta mathrmK$\mathrm{W}_{\text{all}} = \Delta \mathrm{K}$ mathrmM g (mathrmR + mathrmR cos 60^circ) + frac12 mathrmk (mathrmR^2 - 0^2) = frac12 mathrmm v^2$\mathrm{M g} (\mathrm{R} + \mathrm{R} \cos 60^{\circ}) + \frac{1}{2} \mathrm{k} (\mathrm{R}^2 - 0^2) = \frac{1}{2} \mathrm{m v}^2$ Simplifying the gravitational shift and potential expansions: mathrmM g frac3mathrmR2 + fracmathrmkmathrmR^22 = frac12 mathrmm v^2$\mathrm{M g} \frac{3\mathrm{R}}{2} + \frac{\mathrm{k}\mathrm{R}^2}{2} = \frac{1}{2} \mathrm{m v}^2$ ### Step 1: Final Kinematic Value mathrmv = sqrt3mathrmgR + fracmathrmkmathrmR^2mathrmm$\mathrm{v} = \sqrt{3\mathrm{gR} + \frac{\mathrm{k}\mathrm{R}^2}{\mathrm{m}}}$ Matches parameters specified by option (4). ### Pattern Recognition Isolate spring metrics at node points: Initial extension equals 2mathrmR - mathrmR = mathrmR$2\mathrm{R} - \mathrm{R} = \mathrm{R}$. Final extension is 0$0$ since spring length matching mathrmR$\mathrm{R}$ satisfies unextended conditions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Work, Energy and Power

### Core Logic Assertion A: Central force configurations depend solely on the positional distance parameter mathrmr$\mathrm{r}$. They are strictly conservative fields, making path metrics completely irrelevant for total work evaluation. (True) Reason R: Non-conservative profiles like friction or drag dissipate thermal energy paths and do not possess any state potential energy function. (True) Since statement R provides general information about non-conservative forces rather than stating why central fields are path independent, it fails as a direct explanatory bridge. ### Step 1: Final Conclusion Both assertions are factually accurate, but R is not the appropriate explanation for A. This selects option (2). ### Pattern Recognition Conservative fields allow defining potential profiles (F = -nabla mathrmU$F = -\nabla \mathrm{U}$); non-conservative structures completely break this relation. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Work, Energy and Power

### Related Formula Conservation of Mechanical Energy: E_texttotal = K + U = textconstant$E_{\text{total}} = K + U = \text{constant}$ At the initial height S$S$ (velocity v = 0$v = 0$): E_texttotal = mgS$E_{\text{total}} = mgS$ At any height x$x$ above the ground: U = mgx quad textand quad K = frac12mv^2$U = mgx \quad \text{and} \quad K = \frac{1}{2}mv^2$ ### Core Logic Let the height of the particle at that instant be x$x$. We are given: K = 3U$K = 3U$ Substitute the energy terms: frac12mv^2 = 3mgx$\frac{1}{2}mv^2 = 3mgx$ By energy conservation: K + U = E_texttotal$K + U = E_{\text{total}}$ 3U + U = mgS implies 4U = mgS$3U + U = mgS \implies 4U = mgS$ 4(mgx) = mgS implies x = fracS4$4(mgx) = mgS \implies x = \frac{S}{4}$ ### Step 1: Calculating the Speed Now find the speed v$v$ at this height x = S/4$x = S/4$. Since K = 3U$K = 3U$: frac12mv^2 = 3mgx$\frac{1}{2}mv^2 = 3mgx$ frac12mv^2 = 3mgleft(fracS4right)$\frac{1}{2}mv^2 = 3mg\left(\frac{S}{4}\right)$ v^2 = frac6gS4 = frac3gS2$v^2 = \frac{6gS}{4} = \frac{3gS}{2}$ v = sqrtfrac3gS2$v = \sqrt{\frac{3gS}{2}}$ ### Pattern Recognition Standard ratio trick: If K = n U$K = n U$, then by energy conservation (n+1)U = E_texttotal$(n+1)U = E_{\text{total}}$. This immediately yields: x = fracSn+1$x = \frac{S}{n+1}$ Here n = 3$n = 3$, so x = S/4$x = S/4$. This rapid shortcut lets you find the height in a split second! ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Work, Energy and Power

### Core Logic Potential energy functions are strictly mathematically defined exclusively for conservative force interactions via the relationship F = -fracdUdx$F = -\frac{dU}{dx}$[cite: 727]. Coulomb's force, Gravitational force, and Spring restoring force are completely path-independent conservative fields[cite: 146, 147, 149]. Frictional force is a path-dependent, dissipative non-conservative force[cite: 148, 727, 728]. Consequently, it is impossible to define a scalar potential energy function for mechanical friction[cite: 727, 728]. ### Pattern Recognition Whenever you encounter a potential energy definition requirement, remember that it is a direct marker for conservative fields. Dissipative forces like friction or viscous drag instantly break this condition[cite: 727, 728]. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Work, Energy and Power