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Parabola puzzle

Posted on: November 25th, 2014 by
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Let y=ax^2+b_1x+c_1 and y=ax^2+b_2x+c_2 be two parabolas with the same coefficient for the x^2 term. The y-intercepts of the two parabolas are (0,c_1) and (0,c_2) respectively. Let the two parabolas intersect at (x_1,y_1) and let the tangent to the two parabolas drawn at (0,c_1) and (0,c_2) intersect at (x_2,y_2). Show using physics that x_1=x_2.

Hint: Assume x is time and y is space, and there is acceleration due to gravity of a/2 in this 1-d space. View objects first in an inertial frame, and then in an accelerated frame of reference accelerating forward at a/2.

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