Suppose you have a ladder leaning vertically on a wall, and the ladder slips down along the wall to the floor. Consider the point in the center of the ladder. What curve will it draw in the air as the ladder slides down the wall?

As the ladder slides down the wall, the point in the center of the ladder will trace a curve known as a cycloid. A cycloid is a specific type of curve given by the path traced by a point on the circumference of a rolling circle. In this case, the ladder acts as the rolling circle, and the point at the center of the ladder corresponds to the point on the circumference of the rolling circle.

To understand the curve traced by the center of the ladder as it slides down the wall, we can break down the motion into different parts.

First, let's assume that the ladder is initially positioned upright, making a right angle with the floor and the wall. As it starts to slip, the top of the ladder moves downwards along the wall while the bottom moves away from the wall. This creates a rotation around the top point where the ladder is in contact with the wall.

As the ladder continues to slide, the top point moves downward along a straight line perpendicular to the wall, following a vertical path. At the same time, the bottom point moves horizontally along the floor, following a straight line parallel to the wall.

The midpoint of the ladder, being exactly halfway between the top and bottom points, traces out a curve that is a combination of the vertical and horizontal motions of its endpoints. This curve is called a catenary.

A catenary is the shape formed by a hanging chain or cable, and it is described by the mathematical equation y = a*cosh(x/a), where "a" is a constant that determines the shape of the curve.

In the case of the ladder sliding down the wall, the catenary curve traced by the midpoint of the ladder will have a symmetry axis along the straight line that is parallel to the wall and coincides with the bottom point's movement. The shape of the curve will depend on the length of the ladder, the angle at which it starts to slide, and the acceleration due to gravity.

To determine the specific equation of the catenary, you would need to know the length of the ladder and the angle at which it starts sliding. By applying mathematical principles and trigonometry, you can derive the equation for the catenary curve.