Explain acceleration at constant speed

A steady speedometer can accompany a changing velocity

A particle goes counterclockwise around a horizontal circle. At the rightmost point it moves up the page; at the top it moves left; at the leftmost point it moves down; at the bottom it moves right. Here “up the page” is a direction across the horizontal surface, not upward against gravity. We are looking down from above.

Its speed can remain 2 m/s throughout, yet its velocity changes because direction is part of velocity. Acceleration measures change in the velocity vector, not just change in the speedometer reading. Uniform circular motion means constant speed on a circle; it does not mean constant velocity.

Read the four rows of the vector ledger before changing any controls. With radius 2 m, the rightmost position is (2, 0) m and velocity is (0, 2) m/s. At the top, position is (0, 2) m and velocity is (−2, 0) m/s. Rightward is the positive x direction and up the page is the positive y direction. A minus sign identifies the opposite direction along that axis.

The velocity arrows are tangent to the circle: they point along the instantaneous direction of travel and are perpendicular to the radius. A tangent arrow is not a promise that the object continues in a straight line while the inward force remains present. It describes velocity at one instant.

Where the acceleration points

Compare the velocity just before and just after a point on the circle. The small change in velocity points toward the inside of the bend. As the comparison interval becomes very short, that change per unit time points directly toward the center. The instantaneous acceleration is therefore inward, perpendicular to the instantaneous velocity.

The acceleration magnitude is a = v²/r. Here v is nonnegative speed in metres per second, r is the positive circle radius in metres, and a is inward acceleration in metres per second squared. The units agree: (m²/s²)/m = m/s². “Centripetal” is another word for this center-directed acceleration.

For v = 2 m/s and r = 2 m, a = 2²/2 = 2 m/s². At the rightmost point, acceleration is (−2, 0) m/s²; at the top, it is (0, −2) m/s². Its direction changes as the object travels, even though its magnitude remains fixed. Do not apply a single constant world-direction acceleration for an entire revolution.

The displayed acceleration arrow does not mean the particle moves straight toward the center. Its existing tangent velocity and continuously changing inward acceleration combine to bend the path. The diagrams use separate scales for position, velocity and acceleration because these quantities have different units. A velocity arrow and an acceleration arrow of the same drawn length need not have equal numerical values, and neither arrow is a distance traveled.

Period and angular rate answer different questions

A complete circle has length 2πr, where π is approximately 3.14159. At constant nonzero speed, the time for one revolution, called the period T, is distance divided by speed: T = 2πr/v. At 2 m/s and radius 2 m, the circumference is approximately 12.566 m, so the period is approximately 6.283 s. Adjacent rows in the four-position ledger are separated by one quarter of that period.

The angular rate ω, read “omega,” measures angle swept per second. A full revolution is 2π radians. A radian measures arc length divided by radius, so ω = v/r, in radians per second. The default state has angular rate 1 rad/s. This is not one revolution per second: that would require 2π rad/s.

Double speed to 4 m/s at the same 2 m radius. The period halves to approximately 3.142 s, angular rate doubles to 2 rad/s, and inward acceleration becomes 4²/2 = 8 m/s². Acceleration quadruples because speed is squared. Compare the saved states rather than assuming every output doubles when an input doubles.

Treat rest as a separate limiting case

Set speed to zero. The particle remains at the rightmost point, with zero velocity and acceleration. The circle is now a reference set of possible positions, not a path being traveled. The ledger therefore shows only the stationary state; it does not pretend the particle reaches the other three points.

There is no finite revolution period because no revolution is completed. The display reports that condition instead of dividing by zero or printing a period of zero seconds. At small positive speeds, the period becomes large; at exact zero speed, the everyday event “one completed orbit” never occurs in this ideal state.

The OpenStax circular-motion discussion provides background on inward acceleration. This investigation uses original synthetic values, not camera measurements. In real video, a few equally spaced frames can hide changing speed or an imperfect circle. Establish scale, timestamps, measurement uncertainty and enough positions before claiming uniform circular motion.

Transfer: A particle moves counterclockwise at 3 m/s on a circle of radius 3 m. At the leftmost point, identify velocity, acceleration and the revolution period. Is its average velocity over a complete revolution equal to its speed?

Check the circular-motion transfer

Velocity is (0, −3) m/s. Acceleration has magnitude 3²/3 = 3 m/s² and points right, so it is (+3, 0) m/s². The period is 2π × 3 / 3 = 2π s, approximately 6.283 s. Over a complete revolution displacement is zero, so average velocity is zero; average speed remains 3 m/s because the distance traveled is nonzero.