Cosmology. Eternal-universe idea comes full circle

Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is undergoing uniform circular motion. The second, minute, and hour hands of a watch also undergo uniform circular motion. It is hard to believe that points on these rotating objects are actually accelerating, even though the rotation rate is constant. To understand this, we must analyze the motion in terms of...

The Moon orbits around the Earth. In turn, the Earth (and other planets) orbit the Sun. The space directly above our atmosphere is filled with artificial satellites in orbit. One can examine the circular orbit, the simplest kind of orbit, to understand the relationship between the speed and the period of planets and satellites with respect to their positions and the bodies that they orbit.
Nicolaus Copernicus (1473-1543) first suggested that the Earth and all other planets orbit the Sun in...

No object with a finite mass can travel faster than the speed of light in a vacuum. This fact has an interesting consequence in the domain of extremely high gravitational fields.
The minimum speed required to launch a projectile from the surface of an object to which it is gravitationally bound so that it eventually escapes the object’s gravitational field is called the escape velocity. The escape velocity is independent of the mass of the object. Merging the idea of escape velocity with the...

In 1905, Albert Einstein published his special theory of relativity. According to this theory, no matter in the universe can attain a speed greater than the speed of light in a vacuum, which thus serves as the speed limit of the universe.
This has been verified in many experiments. However, space and time are no longer absolute. Two observers moving relative to one another do not agree on the length of objects or the passage of time. The mechanics of objects based on Newton's laws of motion,...

Perceptual constancy is the ability to recognize that objects remain consistent and unchanged even when their appearance varies due to changes in sensory input. There are four main types of perceptual constancy: size constancy, shape constancy, color constancy, and brightness constancy.
Size constancy is the recognition that an object remains the same size, even when its image on the retina changes. For instance, a bus is perceived to be large enough to carry people, even if it looks tiny from...

Indeterminate forms also arise in the evaluation of limits involving products, particularly when one factor approaches zero while the other tends to positive or negative infinity. This situation, commonly described as a zero-times-infinity form, does not have an immediately interpretable outcome. Depending on how the factors behave relative to one another, the limit of such a product may be zero, infinite, or a finite nonzero value.Product Limits and Algebraic RewritingTo analyze limits of this...