Summary Notes: Motion, Speed & Acceleration Vocabulary

Fundamental Concepts of Motion and Physical Quantities

• State of Rest and Motion:

  • At Rest: An object is said to be at rest if it does not change its position over time.
  • In Motion: An object is in motion if it changes its position over time compared to a reference point.

• Major Types of Motion:

  • Linear Motion: Motion along a straight path. Example: An apple falling down.
  • Curvilinear Motion: Motion along a curved path. Example: A rollercoaster travelling over a hill.
  • Circular Motion: Motion around a circular path. Example: The Earth orbiting the Sun.
  • Rotational Motion: Motion around an object's own axis. Example: The Earth spinning on its axis.
  • Oscillatory Motion: Back-and-forth motion around a central position. Example: A clock pendulum.

• Scalar vs. Vector Quantities:

  • Scalar Quantities: Physical quantities that possess only magnitude (size) and lack direction. Examples: Mass, temperature, speed.
  • Vector Quantities: Physical quantities that possess both magnitude (size) and direction. Examples: Force, temperature change, velocity, displacement, acceleration, momentum.

• Distance vs. Displacement:

  • Distance ($d$): The total path length traveled by an object during its motion. It is a scalar quantity with no specified direction.
  • Displacement: The shortest length between an object's starting point and endpoint. It is a vector quantity and includes direction.

Speed Calculations and Formula Triangles

• Definition of Speed ($s$): Speed is defined as the distance an object travels per unit of time.

• Speed Formula and Standard Units:

  • Equation: $s = \frac{d}{t}$
  • Distance ($d$): Measured in meters ($\text{m}$).
  • Time ($t$): Measured in seconds ($\text{s}$).
  • Speed ($s$): Measured in meters per second ($\text{m/s}$).

• Speed Formula Triangle Equations:

  • Formula for Speed: $s = \frac{d}{t}$
  • Formula for Distance: $d = s \times t$
  • Formula for Time: $t = \frac{d}{s}$

• Worked Problem — Speed Calculation:

  • Scenario: A rabbit runs a distance of $100.0\,\text{m}$ in $5.00\,\text{s}$. What is its speed?
  • Formula: $s = \frac{d}{t}$
  • Substitution: $s = \frac{100.0\,\text{m}}{5.00\,\text{s}}$
  • Result: $s = 20.0\,\text{m/s}$

Distance-Time (D-T) Graphs and Types of Speed

• Graph Axes and Slope Characteristics:

  • X-Axis: Time ($t$) is plotted on the horizontal X-axis.
  • Y-Axis: Distance ($d$) is plotted on the vertical Y-axis.
  • Slope: On a distance-time graph, the slope represents the speed of the object.

• Categories of Speed:

  • Average Speed: Speed calculated by dividing the total distance by the total time ($s_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}}$).
  • Instantaneous Speed: Speed measured at an exact moment in time.
  • Constant Speed: Speed that is not changing over time.

• Interpreting Line Shapes on D-T Graphs:
  • Steep Line (Line A): Indicates that the object is moving fast.
  • Gradual Line (Line B): Indicates that the object is moving slow.
  • Horizontal Line (Line C): Indicates that the object is at rest (distance remains constant).
  • Curved Line (Line D): Indicates that the object is accelerating.

Velocity and Momentum

• Velocity ($v$):

  • Definition: Velocity is a measurement that includes the speed of an object and its direction.
  • Classification: Velocity is an example of a vector quantity because it requires a direction.

• Momentum ($p$):

  • Definition: Momentum measures the motion of a mass, calculated as the product of mass and velocity.
  • Equation: $p = m \times v$
  • Mass ($m$): Measured in kilograms ($\text{kg}$).
  • Velocity ($v$): Measured in meters per second ($\text{m/s}$).
  • Momentum ($p$): Measured in kilogram meters per second ($\text{kg}\cdot\text{m/s}$).

• Momentum Formula Triangle Equations:

  • Formula for Momentum: $p = m \times v$
  • Formula for Mass: $m = \frac{p}{v}$
  • Formula for Velocity: $v = \frac{p}{m}$

• Worked Problem — Momentum Calculation:

  • Scenario: A $5.4\,\text{kg}$ bowling ball moves at $8.5\,\text{m/s}$ towards the pins. What is its momentum?
  • Formula: $p = m \times v$
  • Substitution: $p = 5.4\,\text{kg} \times 8.5\,\text{m/s}$
  • Result: $p = 45.9\,\text{kg}\cdot\text{m/s} \text{ (to the pins)}$

Principles of Acceleration

• Definition of Acceleration ($a$): Acceleration is the rate of change of an object's velocity over time.

• Triggers of Acceleration:

  • Acceleration occurs when an object changes its speed, its direction, or both.
  • Like velocity, acceleration is a vector and therefore must include a direction.

• Types of Acceleration:

  • Positive Acceleration ($+$): When speed increases, such as a car accelerating at a green light.
  • Negative Acceleration ($-$): When speed decreases, such as a car slowing down at a red light.
  • Centripetal Acceleration: When direction changes without necessarily changing speed, such as a car turning a corner.
  • Zero Acceleration ($0$): When speed and direction do not change at all.

• Acceleration Formula and Standard Units:

  • Equation: $a = \frac{v_f - v_i}{t}$
  • Velocity ($v$): Measured in meters per second ($\text{m/s}$), where $v_f$ is final velocity and $v_i$ is initial velocity.
  • Time ($t$): Measured in seconds ($\text{s}$).
  • Acceleration ($a$): Measured in meters per second squared ($\text{m/s}^2$).

• Acceleration Formula Triangle Equations:

  • Formula for Acceleration: $a = \frac{v_f - v_i}{t}$
  • Formula for Time: $t = \frac{v_f - v_i}{a}$
  • Formula for Velocity Change: $v_f - v_i = a \times t$

• Worked Problem — Acceleration Calculation:

  • Scenario: A car accelerates from rest ($v_i = 0.00\,\text{m/s}$) to $26.8\,\text{m/s}$ east in $13.4\,\text{s}$. What is the acceleration?
  • Formula: $a = \frac{v_f - v_i}{t}$
  • Substitution: $a = \frac{26.8\,\text{m/s} - 0.00\,\text{m/s}}{13.4\,\text{s}}$
  • Result: $a = (+)\,2.00\,\text{m/s}^2 \text{ (to the east)}$

Graphing Acceleration: Distance-Time and Speed-Time Graphs

• Distance-Time (D-T) Graph Acceleration Profiles:

  • Steep Upward Curve (Curve A): Indicates that the object is $(+)$ accelerating rapidly.
  • Gradual Upward Curve (Curve B): Indicates that the object is $(+)$ accelerating gradually.
  • Steep Downward Curve (Curve C): Indicates that the object is $(-)$ accelerating rapidly.
  • Straight Diagonal Line (Line D): Indicates that the object is moving at a constant speed (no acceleration).

• Speed-Time (S-T) Graph Characteristics:

  • X-Axis: On a speed-time or S-T graph, Time is plotted on the X-axis.
  • Y-Axis: On a speed-time or S-T graph, Speed is plotted on the Y-axis.
  • Slope: On a speed-time or S-T graph, the slope is the acceleration of the object.