Centripetal Acceleration

Acceleration is any change in velocity, and velocity is a vector with both size and direction. A car rounding a curve at a steady 20 m/s never changes speed, but its direction of travel is constantly changing, so the velocity vector is changing, so the car is accelerating. There's no contradiction here: 'constant speed' only promises the magnitude stays fixed, not the direction.

Picture two velocity vectors taken an instant apart as the object sweeps through a small angle. Both have the same length, but they point slightly different ways. Subtract them to get the change in velocity, and that tiny difference vector points inward, toward the center of the circle, not along the path. Dividing by the time interval and shrinking it toward zero gives the instantaneous acceleration, still pointing toward the center, with magnitude a_c = v²/r. This is centripetal acceleration: not a new kind of physics, just the geometric consequence of turning.

Centripetal acceleration needs a cause, and that cause is whatever real force or combination of forces points toward the center at that moment: tension in a string, gravity on an orbiting satellite, friction on a car's tires. That net inward force is what people mean by 'centripetal force,' but it is a label for a role, not a separate push you add to a free-body diagram. This sets up the next step: when an object also speeds up or slows down along the circle, you get a second acceleration component tangent to the path, and the two combine into one total acceleration vector that points somewhere between inward and forward.