if you're standing on ice-skates and throw a basketball forward. how your motion after you throw it compares with the motion of the basketball?

To determine your motion when you throw a basketball while
you are standing on ice-skates; we use the principle of conservation of total
momentum.

Let us assume the
following:

your mass: M

the
mass of the ball: Mb

the velocity of the ball when it is
thrown: Vb in the horizontal direction

your velocity:
V

We have assumed that the ball is thrown in a horizontal
direction. The total momentum of the system which includes you and the ball was 0
initially, after you threw the ball it became Vb*Mb -
V*b

So we arrive at Vb*Mb - V*b =
0

=> V = Vb*Mb / M

This
means you are pushed back with a velocity equal to the product of the mass of the ball
and the velocity of the ball divided by your mass.

The ball
is pulled downward by the gravitation force of attraction equal to Mb*g after it is
released. If it was held at a height of d when it was thrown, the time taken by the ball
to strike the ground is sqrt (2*d/g). During this duration the ball has moved in the
horizontal direction by a distance Vb*sqrt (2*d/g).

You
follow a horizontal path after the ball is released, whereas the ball initially follows
a parabolic path downwards. If we take the force of friction to be negligible, you
continue to move in a straight line, and the ball after it strikes the ground starts to
move in a straight horizontal line too.