8.1 Displacement, Velocity, and Acceleration as Functions of Time
Fundamental Definitions
- Displacement ($s$): The distance of an object from a fixed reference point $O$ in a specified direction.
- $s = 0$: Object is at the fixed point $O$.
- $s > 0$: Object is to the right / positive side of $O$.
- $s < 0$: Object is to the left / negative side of $O$.
- Velocity ($v$): The rate of change of displacement with respect to time ($t$).
- $v = 0$: Object is stationary or instantaneously at rest.
- $v > 0$: Object is moving to the right / in the positive direction.
- $v < 0$: Object is moving to the left / in the negative direction.
- Initial Velocity ($v_0$): Velocity when $t = 0$.
- Maximum / Minimum Velocity: Occurs when acceleration $a = \frac{dv}{dt} = 0$.
- Acceleration ($a$): The rate of change of velocity with respect to time ($t$).
- $a = 0$: Object moves with uniform/constant velocity, or velocity is at a maximum/minimum.
- $a > 0$: Velocity of the object is increasing (acceleration).
- $a < 0$: Velocity of the object is decreasing (deceleration/retardation).
- Initial Acceleration ($a_0$): Acceleration when $t = 0$.
Mathematical Relationships (Calculus Connection)
Movement between displacement $s$, velocity $v$, and acceleration $a$ relies on differentiation and integration with respect to time $t$:
$$\text{Displacement } (s) \xrightarrow[\int s \, dt]{\frac{d}{dt}} \text{Velocity } (v) \xrightarrow[\int v \, dt]{\frac{d}{dt}} \text{Acceleration } (a)$$
- $v = \frac{ds}{dt} \implies s = \int v \, dt$
- $a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \implies v = \int a \, dt$
8.2 Total Distance Traveled
To find the total distance traveled by an object during a time interval $t_1 \le t \le t_2$:
- Find the time(s) $t$ when the object changes direction by setting velocity $v = 0$.
- Calculate the displacement $s$ at $t = t_1$, at each turning point $t = t_{\text{turn}}$, and at $t = t_2$.
- Sum the absolute differences between consecutive displacements:
$$\text{Total Distance} = |s(t_{\text{turn}}) - s(t_1)| + |s(t_2) - s(t_{\text{turn}})|$$
Note: Displacement $\Delta s = s(t_2) - s(t_1)$ represents net position change, whereas Total Distance measures the entire length of the path traveled.