Acceleration measures how quickly an object's velocity changes over time, given by a(t) = dv/dt = d^2x/dt^2. Momentum (p) equals mass (m) times velocity (v), while force (F) equals mass times acceleration, so the derivative of momentum is dp/dt = d(mv)/dt = m(dv/dt) = ma = F. For instance, when you slam on the brakes in a car, the rapid decrease in velocity (acceleration) leads to a force, causing you to feel the deceleration.
It's worth noting that the rate of change of acceleration, known as "jerk," is particularly crucial when planning curves on transportation routes like railways and highways.
In circular motion, there's constant acceleration toward the center of the circle. If a straight road suddenly transitions into a sharply curved section, there's a rapid change in lateral acceleration, resulting in a significant jerk. To mitigate this, curves are designed to start gradually and then gradually tighten, ensuring that the rate of change of curvature remains manageable. This is essential for safe and comfortable travel.