Comprehensive One-Dimensional Kinematics Study Guide

Displacement and Position

• Position: The location of an object relative to a reference frame (e.g., Earth, a whiteboard, or a moving airplane).

• Displacement: The net change in an object's position within a reference frame.

  • Formula: $\Delta x = x_{\text{f}} - x_i$

  • $\Delta$ (Delta): Denotes change in the variable that follows it.

  • $x_{\text{f}}$: Final position.

  • $x_i$: Initial position.

  • SI Unit: Meter ($\text{m}$).

  • Vector Property: Displacement is a vector quantity, possessing both magnitude and direction.

• Distance vs. Displacement:

  • Distance: The magnitude or size of displacement evaluated without direction or sign.

  • Key Difference: Displacement depends strictly on initial and final positions; distance travelled depends on the specific path taken.

  • Example (Cyclist):

  • Travels $3\,\text{km}$ west ($\Delta x_1 = -3\,\text{km}$) and $2\,\text{km}$ east ($\Delta x_2 = +2\,\text{km}$).

  • Total Displacement: $\Delta x = -3\,\text{km} + 2\,\text{km} = -1\,\text{km}$

  • Magnitude of Displacement: $|\Delta x| = |-1\,\text{km}| = 1\,\text{km}$

  • Distance Traveled: $|-3\,\text{km}| + |+2\,\text{km}| = 3\,\text{km} + 2\,\text{km} = 5\,\text{km}$

Vectors, Scalars, and Coordinate Systems

• Vectors:

  • Defined by both magnitude and direction.

  • Examples: Displacement ($-4.0\,\text{m}$), velocity ($90\,\text{km/h}$ east), force ($500\,\text{N}$ downward).

  • Graphical Representation: Drawn as arrows pointing in the direction of motion with length proportional to magnitude.

• Scalars:

  • Defined completely by magnitude alone (no direction).

  • Examples: Temperature ($-10\,^\circ\text{C}$), energy ($250\,\text{kcal}$), speed limit ($90\,\text{km/h}$), height ($1.8\,\text{m}$), distance ($2.0\,\text{m}$).

• Speed vs. Velocity:

  • Speed: Scalar quantity measuring magnitude only.

  • An object moving in a curve at a constant speed has changing velocity because its direction changes.

    • Direction changes - velocity is a vector quantity, so it changes too

Velocity and Speed

Average Velocity ($\bar{v}$):

• Net displacement divided by elapsed time.

• Formula: $\bar{v}=\frac{\Delta x}{\Delta t}=\frac{x_{\text{f}}-x_{i}}{t}$

• Vector quantity sharing the sign of displacement.

• SI Unit: Meters per second ($\text{m/s}$).

• Constant velocity = Flat horizontal line (answer is C)

• In a v(t) graph, the area under the curve = change in displacement

• Negative A = Negative displacement

  

• Instantaneous Velocity ($v$):

  • Velocity at a precise instant over an infinitesimally small time interval ($\Delta t \rightarrow 0$).

  • Instantaneous speed equals the magnitude of instantaneous velocity.

• Average Speed:

  • Total distance travelled divided by elapsed time.

  • Formula: $\text{Average Speed} = \frac{\text{Distance Traveled}}{\text{Elapsed Time}}$

  • Non-directional scalar; not necessarily equal to the magnitude of average velocity.

  • Example (Round Trip): Driving $3\,\text{km}$ to a store and $3\,\text{km}$ back in $0.5\,\text{h}$:

  • Distance Traveled = $6\,\text{km}$

  • Displacement $\Delta x = 0\,\text{km}$

  • Average Speed = $\frac{6\,\text{km}}{0.5\,\text{h}} = 12\,\text{km/h}$

  • Average Velocity $\bar{v} = 0\,\text{km/h}$

Acceleration

• Average Acceleration ($\bar{a}$):

  • The rate at which velocity changes over time.

  • Formula: $\bar{a} = \frac{\Delta v}{\Delta t} = \frac{v_{\text{f}} - v_0}{t_{\text{f}} - t_0}$

  • SI Unit: Meters per second squared ($\text{m/s}^2$).

  • Vector quantity pointing in the direction of velocity change ($\Delta v$).

  • Results from a change in speed magnitude, a change in direction, or both.

• Deceleration vs. Negative Acceleration:

  • Deceleration: Acceleration directed opposite to instantaneous velocity, reducing speed. (Slowing down)

  • Negative Acceleration: Acceleration in the negative coordinate direction ($-x$ or $-y$).

• In this example: This is deceleration and negative acceleration because the runner is slowing down, and because v(t) is positive and the runner is slowing down, the acceleration must point in the negative direction

• Speeding Up vs. Slowing Down:

  • Speeding Up: Velocity and acceleration share the same sign (both positive or both negative).

  • Slowing Down (Deceleration): Velocity and acceleration have opposite signs