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mid-term-sol - phz3113midterm.nb 1 f = Hm v1 ^ 2 m v2 ^ 2L...

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f = H m * v1^2 + m * v2^2 LêH m * va^2 + m * vb^2 L ; g = m * va + m * vb - m * v1 - m * v2; Solve @8 f == 2 ê 3, g == 0 < , 8 v1, v2 <D 99 v1 1 fffff 6 J 3 va + 3 vb è!!! 3 "################################## va 2 6 va vb + vb 2 N , v2 1 6 J 3 va + 3 vb + 3 "################################## va 2 6 va vb + vb 2 N= , 9 v1 1 6 J 3 va + 3 vb + 3 "################################## va 2 6 va vb + vb 2 N , v2 1 6 J 3 va + 3 vb 3 "################################## va 2 6 va vb + vb 2 N== before collision (va-vb)>0, after collision (v1-v2) must be < 0, so first set of solution is un-physical. In addition, va^2-6va vb + vb^2 must be positive, so (va-vb)^2>4 va vb In[1]:= v = kq H 1 ê Sqrt @H x - a L ^2 + H y - a L + z^2 D + 1 ê Sqrt @H x + a L + H y + a L + D - 1 ê Sqrt @H x - a L + H y + a L + D - 1 ê Sqrt @H x + a L + H y - a L + DL Out[1]= kq i k j j j j 1 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! H a + x L 2 + H a + y L 2 + z 2 1 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! H a + x L 2 + H a + y L 2 + z 2 1 è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! H a + x L 2 + H a + y L 2 + z 2 + 1 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! H a + x L 2 + H a + y L 2 + z 2 y { z z z z In[2]:= efield = 8 D @ v, x D , D @ v, y D , D @ v, z D< Out[2]= 9 kq i k j j j j a + x fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + a + x fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + a + x ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 a + x HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 y { z z z z , kq i k j j j j a + y HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + a + y HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + a + y HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 a + y HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 y { z z z z , kq i k j j j j z HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + z HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + z HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 z HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 y { z z z z = In[3]:= firstterm = D @ efield, a D secondterm = D @ D @ efield, a D , a D phz3113midterm.nb 1
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Out[3]= 9 kq i k j j j j 3 H a + x L H 2 H a + x L 2 H a + y LL ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 HH a + x L 2 + H a + y L 2 + z 2 L 5 ê 2 + 1 fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 3 H a + x L H 2 H a + x L 2 H a + y LL fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 HH a + x L 2 + H a + y L 2 + z 2 L 5 ê 2 + 1 fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 3 H a + x L H 2 H a + x L + 2 H a + y LL ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 HH a + x L 2 + H a + y L 2 + z 2 L 5 ê 2 1 HH a + x L 2 + H a + y L 2 + z 2 L 3 ê 2 + 3 H a + x L H 2 H a + x L + 2 H a + y LL fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 HH a + x L 2 + H a + y L 2 + z 2 L 5
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