The concept of mass increase with speed is an outdated and incorrect idea. According to modern physics, an object's mass does not increase as its speed increases. Instead, its relativistic mass was once used to describe the phenomenon of increased momentum with velocity, but this concept is no longer favored.
In modern physics, the mass of an object is considered an invariant quantity known as its rest mass or invariant mass. This rest mass remains constant regardless of the object's speed. It is an inherent property of the object that does not change with its motion.
On the other hand, weight is the force experienced by an object due to gravity. Weight is dependent on the gravitational field strength and the mass of the object. When we say an object's weight remains constant, it means that the force of gravity acting on the object is constant. Weight does not depend on the object's speed but on its mass.
So, to summarize, the mass of an object does not increase with speed, but its weight remains constant because weight is determined by the object's mass and the gravitational force acting on it.