You're interested in using mechanical energy to explore the motion of a ball you throw up in the air. Where do you need to put the origin of your y-axis?

When studying the motion of a ball thrown up in the air using mechanical energy, you would typically place the origin of your y-axis at the point from which the ball is thrown. This is because the motion and energy changes of the ball are most conveniently described relative to this point.

Here are the steps:

1. Identify the starting point of the ball's motion. This is where the ball is initially thrown from.

2. Set this point as the origin (0,0) of your y-axis.

3. The positive direction of the y-axis is typically upwards, following the direction of the ball's motion.

4. Any points above the origin along the path of the ball will have positive y-values, while any points below will have negative y-values.

5. Now you can track the ball's motion and calculate its mechanical energy at different points along the y-axis.

Remember, the choice of origin is somewhat arbitrary and can be chosen for convenience. The physical laws describing the ball's motion will be the same regardless of where you place the origin.