Acceleration (differential geometry)

Okay, let's imagine that you are a little car riding down a bumpy road. As you go along, you might feel yourself speeding up or slowing down, right? Well, acceleration is like the way your speed changes over time.

In fancy mathematical terms, acceleration is the rate at which your velocity (which is a fancy word for your speed and the direction you're going in) changes. So if you're going really fast in a straight line, your velocity isn't changing and your acceleration is zero. But if you start to turn or stop suddenly, your velocity is changing and your acceleration is not zero.

Now, let's imagine driving our little car on a curved road. If we're taking a tight turn, we might feel our body push against the car door, right? This is because our acceleration is not as simple as just speeding up or slowing down - it's also pulling us in a different direction.

This kind of acceleration is called "tangential acceleration" because it's happening along the curve of the road. Another kind of acceleration that can happen on a curved road is "normal acceleration", which is what happens when the road curves so much that our car feels like it's being pushed off the road.

So, to sum up: acceleration is the rate at which our velocity (our speed and the direction we're going in) changes. When we're on a curved path, we might feel two different kinds of acceleration - tangential acceleration, which is pulling us along the curve, and normal acceleration, which is pushing us away from the curve.