The second derivative of distance with respect to time is equal to acceleration because acceleration is defined as the rate of change of velocity with respect to time, and velocity is the first derivative of distance with respect to time.
Let's break it down step by step:
Distance: Distance (s) is a measure of how far an object has traveled. It is typically denoted as a function of time, s(t), where t represents time.
Velocity: Velocity (v) is defined as the rate of change of distance with respect to time. Mathematically, it is the derivative of distance with respect to time, denoted as v(t) = ds(t)/dt.
Acceleration: Acceleration (a) is defined as the rate of change of velocity with respect to time. Mathematically, it is the derivative of velocity with respect to time, denoted as a(t) = dv(t)/dt.
Second derivative: Taking the derivative of velocity with respect to time gives the second derivative of distance. Mathematically, it is denoted as d²s(t)/dt², which represents the rate of change of acceleration with respect to time.
By substituting the definition of velocity (v(t) = ds(t)/dt) into the equation for acceleration (a(t) = dv(t)/dt), we can rewrite the equation as a(t) = d²s(t)/dt².
Therefore, the second derivative of distance with respect to time is equal to acceleration. This relationship allows us to analyze the motion of objects by examining how acceleration, velocity, and distance change with respect to time.