Is it possible for an object to have a constant speed but be accelerating?

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Yes.  'Acceleration' does NOT mean speeding up. Acceleration means any change in the speed or direction of motion. Speeding up, slowing down, or rounding a curve are all accelerations. Any object that's not moving in a straight line is accelerating, even if its speed is constant.   -- Your car is accelerating when you drive 30 mph along a curved road. -- You're accelerating on a merry-go-round (carousel) or a ferris wheel when it's speeding up, slowing down, or turning at a constant rate. -- Your car is accelerating when you see a red light and step on the brakes. -- You are accelerating when you jog around a circular track at gym.

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