problem07_87

University Physics with Modern Physics with Mastering Physics (11th Edition)

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7.87: a) Eliminating β in favor of α and ) ( 0 0 x β x α = , . ) ( 0 2 0 2 0 0 2 2 0 2 0 2 - = - = - = x x x x x x x x x x x x x U α α α β α 0 ) 1 1 ( ) ( 2 0 0 = - = x x U α . ) ( x U is positive for 0 x x < and negative for 0 x x ( α and β must be taken as positive). b) . 2 2 ) ( 2 0 0 2 0 - = - = x x x x mx U m x v α The proton moves in the positive x -direction, speeding up until it reaches a maximum speed (see part (c)), and then slows down, although it never stops. The minus sign in the square root in the expression for ) ( x v indicates that the particle will be found only in the region where 0 < U , that is, 0 x x . c) The maximum speed corresponds to the maximum kinetic energy, and hence the minimum potential energy. This minimum occurs when 0 = dx dU , or , 0 2 3 2 0 3 0 0 = + - = x x x x x dx dU α which has the solution 0 2 x x = .
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