Curved Path Dynamics: Essentials of Velocity, Speed, and Acceleration
Velocity along a curve is the derivative of a position vector function \(\mathbf{r}(t)\), providing direction and magnitude. Speed is the scalar magnitude of velocity, calculated as \(|\mathbf{r}'(t)|\). Acceleration is the derivative of velocity, \(\mathbf{r}”(t)\), indicating changes in velocity’s direction or speed. In curved motion, acceleration has tangential (speed changes) and normal (directional changes) components. These […]
Average Value of a Curve
The average value of a function over an interval \( [a, b] \) is computed as the integral of the function over that interval, divided by the interval’s length. This yields the function’s mean value, representing its average height on the interval \( [a, b] \).