showing equivialance of two versions of angular momentum equations by Vector math

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4U mechanics question - find: maximum speed attained; distance travelled

Please help i have trouble understand this problem and i could not apply appropriate formulae to solve it. I had tried but still obtain the correct answer. The problem is: A particle of mass 10kg, moving in a straight line, starts at rest from a point A under the action of a force that decreases uniformly from 20N to zero in 20 secs. IT then travels with a constant speed for a further 20s, and finally moves under the action of an opposing force of 40N until it comes to rest at B. Find the maximum speed attained and the distance travelled.

Simply the following expression as much as possible.

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Using direct tensor notation prove the following.

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A tensor transforms any vector:

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