How does centripetal acceleration differ from linear acceleration?
The direction of course...
Centripetal acceleration is the acceleration of an object traveling in a circle, while linear acceleration is the acceleration of an object traveling in a straight line.
In calculus terms, centripetal acceleration is the rate of change of tangential velocity, that is the velocity of an object measured at any point tangent to a circle. Meanwhile linear acceleration is the rate of change of linear velocity, which is the velocity of an object traveling in a straight line.
Linear acceleration causes speed to increase or decrease. Centripetal acceleration does not change speed but changes direction of the velocity so the object curves.
Consider a small satellite that is about to be launched. If a workman mounting it on top of the rocket drops it, is would fall to the ground and experience linear acceleration on the way down. When it lands, there would be linear acceleration in the opposite direction causing it to slow down. Linear acceleration causes an object's speed to increase or decrease.
On the other hand, assume the satellite is launched successfully into a circular orbit. The only acceleration it would experience once it is placed in orbit would be centripetal. The word centripetal means toward the center. The direction of centripetal acceleration is toward the center of the circular orbit. Therefore centripetal acceleration does not cause the speed to increase. It causes the direction of the velocity vector to continually change so that the satellite goes in a circle.
I hope this helps,