Kinematics: Displacement, Velocity, and Acceleration (Page 1 notes)
Displacement vs Distance Traveled
- Displacement (vector): straight-line vector from initial to final position \Delta \mathbf{r} = \mathbf{r}{\text{final}} - \mathbf{r}{\text{initial}} with magnitude |\Delta \mathbf{r}| .
- Distance traveled (scalar): total length of the path taken, s \ge |\Delta \mathbf{r}|.
- Equality (s = |\Delta \mathbf{r}|) holds only for straight-line motion in a single direction without reversals.
Average vs Instantaneous Velocity
- Average velocity: \bar{v} = \frac{\Delta x}{\Delta t}.
- Instantaneous velocity: v(t) = \frac{dx}{dt}, velocity at a specific instant.
- Relationship: \bar{v} = v_{\text{inst}} if and only if velocity is constant over the interval.
Acceleration
- Constant velocity: If velocity is constant, acceleration a = \frac{dv}{dt} = 0.
- Velocity zero, acceleration nonzero: Velocity can be zero at an instant while acceleration is nonzero (e.g., object starting from rest or at the peak of vertical motion).
- Changing direction with constant acceleration: Yes, velocity can change direction with constant acceleration (e.g., a ball thrown straight up).
Constant Acceleration (a constant)
- In equal time intervals, the change in velocity is the same: \Delta v = a \Delta t. This means the velocity-time graph is a straight line.
Free Fall (absence of friction)
- Acceleration is constant downward: a = g \approx 9.81\ \text{m/s}^2. Velocity increases linearly with time.
Ball Thrown Straight Up: Highest Point
- At the highest point, velocity is zero (v = 0), but acceleration remains downward and is not zero (a = -g).
- Velocity: v = u + a t
- Position: s = u t + \tfrac{1}{2} a t^2
- Velocity-Displacement: v^2 = u^2 + 2 a s
- Displacement: vector; Distance traveled: scalar.
Connections
- Kinematics focuses on motion description: displacement/distance, velocity, acceleration.
- Relevant for predicting motion in various scenarios, including gravitational fields.
