Section 1.5: Derivatives of Position
This section introduces the derivative concepts in kinematics: velocity as the derivative of position, and acceleration as the derivative of velocity. We also practice interpreting motion from graphs.
Velocity:
Instantaneous velocity: \( \vec{v}(t) = \frac{d\vec{r}}{dt} \)
Components: \( v_x = \frac{dx}{dt}, \quad v_y = \frac{dy}{dt} \)
Acceleration:
Instantaneous acceleration: \( \vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2 \vec{r}}{dt^2} \)
Components: \( a_x = \frac{d v_x}{dt}, \quad a_y = \frac{d v_y}{dt} \)
Example 1
Position: \( x(t) = 5t^2 \). Find velocity and acceleration as functions of time.
Velocity: \( v(t) = \frac{dx}{dt} = 10t \)
Acceleration: \( a(t) = \frac{dv}{dt} = 10 \text{ m/s²} \)
Practice Problems
- Given \( x(t) = 3t^3 \), find \( v(t) \) and \( a(t) \).
- Position \( y(t) = 4t - 2t^2 \). Find velocity and acceleration.
- A particle moves along x-axis: \( x(t) = 2t^2 - t \). Compute \( v(t) \) and \( a(t) \).
- For \( r(t) = (t^2, t^3) \), find \( \vec{v}(t) \) and \( \vec{a}(t) \).
- Position function \( x(t) = 6 - 2t^2 \). Determine \( v(t) \) and \( a(t) \).