normal and tangential acceleration

Move the gold point along the curved path with the slider, and the blue velocity arrow hugs the path at every spot, pointing along the tangent. Where the path bends up the arrow points up, where it dips the arrow points down; it always faces the way the point is going and never cuts across the path or looks backward. It sounds obvious, but it is where curvilinear motion starts. Velocity tells which way the position is moving, and a point sliding along a path moves at each instant in the direction the path runs. So going around a curve changes the velocity's direction moment by moment, and something must be causing that change.

The first thing an acceleration can do is change the speed. The figure freezes one instant at one spot on an arc, and the slider sets the tangential acceleration a_t, the acceleration along the path. With positive a_t the acceleration arrow points the same way as the blue velocity v and the readout says speeding up; with negative a_t it points the other way and the readout says slowing down. Its size is a_t = dv/dt, the rate of change of speed from the last lessons. What matters is that the tangential acceleration has nothing to do with direction; it only raises or lowers the speed. Set a_t to 0 and the speed is constant, but on a curve the direction is still changing, so some acceleration remains. The next figure takes up that remaining part.

Now hold the speed fixed and look at the acceleration that only changes direction. A point on the arc keeps changing the direction of its velocity even at constant speed, so it accelerates, and this acceleration points perpendicular to the path, toward the inside of the curve, the center of curvature; it is called the normal acceleration. Its size is a_n = v²ρ, where ρ is the radius of curvature, the radius of the circle that best fits the curve at that spot. Drag the speed slider v to double the speed and the readout's a_n, the size of the orange inward arrow, jumps fourfold, because the speed v enters squared. The more sharply the curve bends, that is, the smaller ρ, the larger a_n becomes at the same speed, so a tighter bend demands a stronger inward pull.

The cleanest case is uniform circular motion. Going around a circle at constant speed, the tangential acceleration is 0 and only the normal part is left; because it points to the center, it is called centripetal acceleration. Turn the point once around the circle with the position slider: the blue velocity arrow stays tangent and turns with it, but the orange acceleration arrow points straight at the center wherever the point is, and its size stays fixed at v²r. You might ask how there can be acceleration when the speed does not change. Acceleration is a change of velocity, not of speed, and a change of direction alone changes the velocity. The gravity that holds the Moon, a stone whirled on a string, and a spinning ride are all explained by this centripetal acceleration.

Now put the two together. The actual acceleration of an object running along a curve is the vector sum of its tangential and normal parts, and switching among three cases with the buttons makes the difference plain. At constant speed the tangential part is 0, so the gold total acceleration points straight inward. When speeding up, the tangential part adds forward and tilts the total toward the direction of motion; when slowing down, it points backward and tilts the total back. In every case the orange inward normal part a_n = v²ρ never disappears as long as the path curves, and the total magnitude adds the two at right angles, a = √(a_t² + a_n²). In the end, understanding acceleration on a curve means splitting it into a part along the path and a part across it.