| The boundary condition will be the same as in the previous problem, but the left side of the inequality will change. | | The boundary condition will be the same as in the previous problem, but the left side of the inequality will change. |
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| The velocity of the center of mass of the system: | | The velocity of the center of mass of the system: |
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| $$\upsilon_C = \frac{m_1\upsilon_1 + m_2\upsilon_2}{m_1 + m_2}$$ | | $$\upsilon_C = \frac{m_1\upsilon_1 + m_2\upsilon_2}{m_1 + m_2}$$ |
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| Hence the velocity of our projectiles: | | Hence the velocity of our projectiles: |
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| $$\upsilon_{1_C} = \frac{m_2(\upsilon_1 - \upsilon_2)}{m_1 + m_2}$$ | | $$\upsilon_{1_C} = \frac{m_2(\upsilon_1 - \upsilon_2)}{m_1 + m_2}$$ |
| $$\upsilon_{2_C} = -\frac{m_1(\upsilon_1 - \upsilon_2)}{m_1 + m_2}$$ | | $$\upsilon_{2_C} = -\frac{m_1(\upsilon_1 - \upsilon_2)}{m_1 + m_2}$$ |
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| Now let's calculate the total kinetic energy in the SCM, it will go to the left side of the inequality: | | Now let's calculate the total kinetic energy in the SCM, it will go to the left side of the inequality: |
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| $$E_K = \frac{(\upsilon_1 - \upsilon_2)^2}{2(\frac{1}{m_1}+\frac{1}{m_2})}$$ | | $$E_K = \frac{(\upsilon_1 - \upsilon_2)^2}{2(\frac{1}{m_1}+\frac{1}{m_2})}$$ |
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| Finite velocities in the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} < \frac{2B^2 l S^2(3s - 2S)}{\mu_0 (S - s)(2S - s)}$： | | Finite velocities in the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} < \frac{2B^2 l S^2(3s - 2S)}{\mu_0 (S - s)(2S - s)}$： |
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| They are obtained by changing the sign of the speeds in the CM system and adding its speed: | | They are obtained by changing the sign of the speeds in the CM system and adding its speed: |
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| $${\upsilon_1}' = \frac{2m_2\upsilon_2 + (m_1 - m_2)\upsilon_1}{m_1 + m_2}$$ | | $${\upsilon_1}' = \frac{2m_2\upsilon_2 + (m_1 - m_2)\upsilon_1}{m_1 + m_2}$$ |
| $${\upsilon_2}' = \frac{2m_1\upsilon_1 + (m_2 - m_1)\upsilon_2}{m_1 + m_2}$$ | | $${\upsilon_2}' = \frac{2m_1\upsilon_1 + (m_2 - m_1)\upsilon_2}{m_1 + m_2}$$ |
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| For the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} > \frac{2B^2 l S^2(3s - 2S)}{\mu_0(S - s)(2S - s)}$ speeds will remain the same: | | For the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} > \frac{2B^2 l S^2(3s - 2S)}{\mu_0(S - s)(2S - s)}$ speeds will remain the same: |
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| $${\upsilon_1}' = \upsilon_1$$ | | $${\upsilon_1}' = \upsilon_1$$ |
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| $${\upsilon_2}' = \upsilon_2$$ | | $${\upsilon_2}' = \upsilon_2$$ |
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| #### Answer | | #### Answer |
| For: | | For: |
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| $$\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} < \frac{2B^2 l S^2(3s - 2S)}{\mu_0 (S - s)(2S - s)}$$ | | $$\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} < \frac{2B^2 l S^2(3s - 2S)}{\mu_0 (S - s)(2S - s)}$$ |
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| $${\upsilon_1}' = \frac{2m_2\upsilon_2 + (m_1 - m_2)\upsilon_1}{m_1 + m_2}$$ | | $${\upsilon_1}' = \frac{2m_2\upsilon_2 + (m_1 - m_2)\upsilon_1}{m_1 + m_2}$$ |
| $${\upsilon_2}' = \frac{2m_1\upsilon_1 + (m_2 - m_1)\upsilon_2}{m_1 + m_2}$$ | | $${\upsilon_2}' = \frac{2m_1\upsilon_1 + (m_2 - m_1)\upsilon_2}{m_1 + m_2}$$ |
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| For the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} > \frac{2B^2 l S^2(3s - 2S)}{\mu_0(S - s)(2S - s)}$ speeds will remain the same: | | For the case of $\frac{(\upsilon_1 - \upsilon_2)^2}{(\frac{1}{m_1}+\frac{1}{m_2})} > \frac{2B^2 l S^2(3s - 2S)}{\mu_0(S - s)(2S - s)}$ speeds will remain the same: |
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| $${\upsilon_1}' = \upsilon_1$$ | | $${\upsilon_1}' = \upsilon_1$$ |
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| $${\upsilon_2}' = \upsilon_2$$ | | $${\upsilon_2}' = \upsilon_2$$ |
