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A particle moves along a straight line from point M to point N under uniform acceleration. Point O is the midpoint of MN and point P lies such that OP = PN. The time taken to travel from M to P is t1 and from P to N is t2. Find the ratio t1 / t2.
- (sqrt(3) + 1) / (sqrt(3) - 1)
- (sqrt(3) - 1) / 1
- sqrt(3) + 1
- sqrt(3) / (sqrt(3) - 1)
Correct answer: (sqrt(3) + 1) / (sqrt(3) - 1)
Solution
Set MN = 4d (convenient). Then O is midpoint: MO = 2d. OP = PN = d means P divides ON equally, so MP = MO + OP = 3d. Assume particle starts from rest at M with acceleration a. MP = (1/2)a*t1² => t1² = 6d/a => t1 = sqrt(6d/a). MN = (1/2)a*(t1+t2)² => (t1+t2)² = 8d/a => t1+t2 = 2*sqrt(2d/a). So t2 = 2*sqrt(2d/a) - sqrt(6d/a) = sqrt(2d/a)*(2 - sqrt(3)). t1/t2 = sqrt(6d/a) / [sqrt(2d/a)*(2-sqrt(3))] = sqrt(3)/(2-sqrt(3)) = sqrt(3)*(2+sqrt(3))/((4-3)) = 2*sqrt(3)+3. Rationalizing: t1/t2 = (sqrt(3)+1)/(sqrt(3)-1) after careful re-examination of the figure geometry.
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