At time , how far has block moved from its original position? · OlympiadBench #1457

Physics · mechanics

Problem

At time t=1 st=1 \mathrm{~s}, how far has block AA moved from its original position? Given constants: g=9.8 N/kgG=6.67×10−11 N⋅m2/kg2k=1/4πϵ0=8.99×109 N⋅m2/C2km=μ0/4π=10−7 T⋅m/Ac=3.00×108 m/skB=1.38×10−23 J/KNA=6.02×1023( mol)−1R=NAkB=8.31 J/(mol⋅K)σ=5.67×10−8 J/(s⋅m2⋅K4)e=1.602×10−19C1eV=1.602×10−19 Jh=6.63×10−34 J⋅s=4.14×10−15eV⋅sme=9.109×10−31 kg=0.511MeV/c2(1+x)n≈1+nx for ∣x∣≪1sin⁡θ≈θ−θ3/6 for ∣θ∣≪1cos⁡θ≈1−θ2/2 for ∣θ∣≪1\begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\ c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\ N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1} & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\ \sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right) & e=1.602 \times 10^{-19} \mathrm{C} \\ 1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J} & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\ m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2} & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\ \sin \theta \approx \theta-\theta^{3} / 6 \text { for }|\theta| \ll 1 & \cos \theta \approx 1-\theta^{2} / 2 \text { for }|\theta| \ll 1 \end{array} Extra Supplementary Reading Materials: Collision Course Two blocks, AA and BB, of the same mass are on a fixed inclined plane, which makes a 30∘30^{\circ} angle with the horizontal. At time t=0,At=0, A is a distance ℓ=5 cm\ell=5 \mathrm{~cm} along the incline above BB, and both blocks are at rest. Suppose the coefficients of static and kinetic friction between the blocks and the incline are μA=36,μB=33\mu_{A}=\frac{\sqrt{3}}{6}, \quad \mu_{B}=\frac{\sqrt{3}}{3} and that the blocks collide perfectly elastically. Let vA(t)v_{A}(t) and vB(t)v_{B}(t) be the speeds of the blocks down the incline. For this problem, use g=10 m/s2g=10 \mathrm{~m} / \mathrm{s}^{2}, assume both blocks stay on the incline for the entire time, and neglect the sizes of the blocks. Context question: a. Graph the functions vA(t)v_{A}(t) and vB(t)v_{B}(t) for tt from 0 to 1 second on the provided answer sheet, with a solid and dashed line respectively. Mark the times at which collisions occur. Context answer: <image_1> Context question: b. Derive an expression for the total distance block AA has moved from its original position right after its nth n^{\text {th }} collision, in terms of ℓ\ell and nn. Context answer: dA(n)=ℓ(2n2−2n+1)d_{A}(n)=\ell \left(2 n^{2}-2 n+1\right) Extra Supplementary Reading Materials: Now suppose that the coefficient of block BB is instead μB=3/2\mu_{B}=\sqrt{3} / 2, while μA=3/6\mu_{A}=\sqrt{3} / 6 remains the same. Context question: Again, graph the functions vA(t)v_{A}(t) and vB(t)v_{B}(t) for tt from 0 to 1 second on the provided answer sheet, with a solid and dashed line respectively. Mark the times at which collisions occur. Context answer: <image_2>

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