What is the relationship between distance, velocity, and acceleration?

The relationship between distance, velocity, and acceleration is rooted in calculus, particularly in the concepts of differentiation.

Distance is often represented as a function of time, which is denoted as d(t). This function gives the total length traveled over time.

Velocity, which is the rate of change of distance with respect to time, is expressed as the first derivative of the distance function. This is represented by d'(t), indicating how quickly the position is changing at any given point in time.

Acceleration, defined as the rate of change of velocity with respect to time, is represented as the second derivative of the distance function or the first derivative of the velocity function. This is denoted by d''(t), reflecting how velocity changes over time.

Therefore, the correct answer encompasses all three relationships: distance is given by d(t), velocity by its first derivative d'(t), and acceleration by the second derivative d''(t). This clear hierarchy illustrates how these three concepts are interconnected.