tangential acceleration
When you round a bend and press the accelerator at the same time, you are speeding up while turning. The part of your acceleration that lies along your direction of travel — the part that actually changes how fast you are going — is the tangential acceleration. It runs along the tangent to the circle, in step with the velocity, quite separate from the inward centripetal part that merely bends your path.
Precisely, for something on a circle of radius r whose spin rate is changing, the tangential acceleration is a_t = r alpha, where alpha is the angular acceleration. It points along the direction of motion (forward when speeding up, backward when slowing). In general circular motion the total acceleration is the vector sum of two perpendicular pieces: the tangential part a_t, which changes the speed, and the centripetal part a_c = v^2/r, which changes the direction. Their combined size is a = sqrt(a_t^2 + a_c^2).
Thinking of these two separately is a powerful habit. Any curved motion at all can be split into 'how fast am I speeding up or slowing?' (tangential) and 'how sharply am I turning?' (centripetal), and the two never interfere with each other.
A wheel of radius 0.4 m has angular acceleration alpha = 5 rad/s^2. A point on its rim has tangential acceleration a_t = r alpha = 0.4 x 5 = 2 m/s^2 along its direction of motion.
Tangential changes the speed; centripetal changes the direction; together they make the full acceleration.
In uniform (constant-speed) circular motion the tangential acceleration is zero, leaving only the centripetal part; tangential acceleration appears only when the spin rate itself changes.