You are standing on a skateboard, initially at rest. a friend throws a very heavy ball towards you. you can either catch the object or deflect the object back towards your friend (such that it moves away from you with the same speed as it was originally thrown). what should you do in order to minimize your speed on the skateboard?

Catch the ball.

Explanation:

In order to MINIMIZE your speed on the skateboard you have to catch the ball. Since momentum is conserved, if you deflect the object back towards your friend now there will be momentum on that direction, and that must be compensated with momentum on the original direction (of the ball thrown), which will be possible only by increasing your speed in that original direction.

find the sum of the two horizontal vectors and the sum of the two vertical vectors

isoceles right angled triangle