An object moving in space will move in a straight line at a constant velocity by Newton's first law of inertia. If the object is constrained in someway not to move in a straight line, then it must be experiencing an acceleration. By definition acceleration is a time rate change of velocity. If you don't want a mathematical proof of this, you must accept intuitively that this acceleration is not proportional to the velocity of the object, but is proportional to the velocity squared. Also it is inversely proportional to the radius of curvature of the object as it curves through space. Remember by definition acceleration is a time rate change of velocity. Velocity is a vector, so it has both magnitude and direction. If the object's direction is changing, then it must be accelerating also, even if the magnitude of the velocity is not. Imagine a car going around a circular race track with a radius of 100 feet at 65 miles per hour. It will experience an acceleration proportional to its speed squared and inversely proportional to its radius. The driver would experience an appreciable centrifugal* force pulling him radially outward. Now imagine the same car going around a circular race track with the same speed, but the radius is now 100 miles. The acceleration will now be less. The driver would barely perceive the centrifugal force in his frame pulling him radially outward. It all boils down to the speed squared and how fast the car is turning in inertial space. Nature does not like change. For some unknown reason that is still not fully understood, inertia rises whenever there is a change in the velocity of an object. No one has yet come up with a fully accepted, bona fide explanation of the cause of inertial forces.
*centrifugal force
Definition/Summary
A non-inertial observer measures the same "real" (physical) forces as an inertial observer does, but if he wants to apply the inertial laws of motion, he must add "fictitious" (non-physical) forces.
One of these "fictitious" forces is a position-dependent centrifugal force (), which must be used by any rotating observer.
For a body stationary relative to a uniformly rotating observer, it is the only "fictitious" force; for a relatively moving body, there is also a velocity-dependent Coriolis force; for a non-uniform rotating observer, there is also a position-dependent Euler force.
These "fictitious" forces are confusingly also called inertial forces (even though they only appear in non-inertial frames) because they are proportional to the mass (the "inertia") of the body.
(By comparison, centripetal force is a "real" force. It is not a separate force, it is another name for an existing physical force, such as tension or friction, which makes a body move in a circle.)
Equations
CENTRIFUGAL FORCE (position-dependent and radially outward) at distance from axis of rotation of an observer with instantaneous angular speed :
CORIOLIS FORCE (velocity-dependent and "magnetic") on velocity relative to the rotating frame of an observer with instantaneous angular velocity :
EULER FORCE (position-dependent and tangential) at distance from axis of angular acceleration of an observer with instantaneous angular acceleration :
Other forces at play indoors that may prevent "running inside the points of measurements but still outside the curb line" idea.
Calling for Dr. Adrian C. Gately to set us straight.