Acceleration in Curved Motion Splits into Tangential and Normal

Move the point along the curved path with the slider. At any position, the velocity arrow hugs the path and points along the tangent. Where the path bends up the arrow turns up, where it dips down the arrow dips down, always facing the way of travel. It never points across the path or backward. It seems obvious, but this is the starting point of curved motion: velocity tells you which way the position is moving, and a point sliding along a path moves, at that instant, in the direction the path runs. So going around a curve turns the velocity's direction at every moment, and something must be causing that change of direction.

The first thing acceleration can do is change the speed. Adjust the tangential acceleration a_t with the slider. When a_t points the same way as the velocity (forward), the point speeds up; when it points the opposite way (backward), it slows down. This acceleration always acts along the path, that is, along the tangent, which is why it is called tangential. Its size is a_t = dv/dt, exactly the rate of change of speed from the last lessons. The key point is that tangential acceleration has nothing to do with direction; it only raises or lowers the speed. Set a_t to zero and the speed becomes constant. Yet on a curve, even at constant speed, an acceleration still remains, because the direction is changing. We look at that leftover in the next scene.

Now look at the acceleration that keeps the speed fixed and only changes direction. The point goes around an arc at constant speed, yet because the velocity's direction keeps turning, there is an acceleration. This acceleration is perpendicular to the path, pointing into the inside of the bend, toward the center of curvature, which is why it is called normal acceleration. Its size is a_n = v²ρ, where ρ is the radius of the circle the curve traces at that point, the radius of curvature. Drag the v slider. Double the speed and a_n jumps fourfold, because v enters as a square. And the tighter the turn (the smaller ρ), the larger a_n. At the same speed, a sharper curve demands a stronger inward pull.

The cleanest case is uniform circular motion: going around a circle at constant speed. Since the speed is constant, the tangential acceleration is zero, and all that remains is the normal acceleration. This one gets a special name, centripetal acceleration, meaning center-seeking. Drag the point around the circle. The velocity arrow always stays tangent and spins around, but the acceleration arrow, at every position, points straight at the center of the circle, unchanged. Its size, too, is constant at v²r. It can sound strange: the speed does not change, so why is there acceleration? Because acceleration is the change of velocity, not of speed, and changing direction alone already changes the velocity. The gravity that holds the Moon, a stone on a string, a spinning ride, are all explained by this centripetal acceleration.

Now put the two together. The true acceleration of an object running along a curve is the vector sum of the tangential and the normal acceleration. Switch the three cases with the buttons. At constant speed the tangential part is zero, so the acceleration points purely inward. While speeding up, the tangential part adds forward, and the total acceleration tilts toward the direction of travel. While slowing down, the tangential part points backward, and the acceleration tilts back. In every case the inward normal part a_n = v²ρ stays alive as long as the path curves. The total size is the two added at right angles, a = √(a_t² + a_n²). In the end, understanding acceleration in curved motion is just splitting it into a part along the path and a part across it. In the next lesson we move to a rotating rigid body and tell the same story again with angular velocity ω and angular acceleration α.