SOLUTION: A 2.70 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as F = 3x^2 - 2x. If the block's speed passin

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: A 2.70 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as F = 3x^2 - 2x. If the block's speed passin      Log On


   

Question 1056854: A 2.70 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as F = 3x^2 - 2x. If the block's speed passing through the origin was 2.80 m/s, with what speed (in meters/second) does it pass the point x = 8.00 m?
Answer by ikleyn(52832) About Me  (Show Source):
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A 2.70 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies
with position as F = 3x^2 - 2x. If the block's speed passing through the origin was 2.80 m/s, with what speed (in meters/second)
does it pass the point x = 8.00 m?
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The Energy Conservation Law says:

The change of the kinetic energy is equal to the work done by the force on the given space interval (distance).

In other words,

%28mv%5B2%5D%5E2%29%2F2 - %28mv%5B1%5D%5E2%29%2F2 = Integral (F(x)dx) from 0 to x = 8 m.

Or, substituting the data

%282.7%2Av%5E2%29%2F2 - %282.7%2A2.8%5E2%29%2F2 = Integral ((3x^2-2x)dx) from 0 to 8,

or

%282.7%2Av%5E2%29%2F2+-+%282.7%2A2.8%5E2%29%2F2 = I(8) - I(0),

where I(x) = x^3 - x^2 is the (indefinite) integral of (3x^2-2x)dx.

Which is 

%282.7%2Av%5E2%29%2F2+-+%282.7%2A2.8%5E2%29%2F2 = 8^3 - 8^2,  or

%282.7%2Av%5E2%29%2F2+-+%282.7%2A2.8%5E2%29%2F2 = 48.

Solve for "v", the velocity under the question.