Kinematics in One Dimension

Kinematics in One Dimension

Introduction to Kinematics

• Kinematics and Dynamics together form the branch of physics known as Mechanics.
• In Kinematics, we treat objects as particles, which are idealized points without size or structure, allowing us to focus solely on their translational motion.
• We typically consider motion along a single straight line, often designated as the x-axis.
• The indices $i$ or $o$ refer to initial values, and $f$ or no subscript refers to final values.

Position and Displacement

• Initial Position: $x_o$
• Final Position: $x$
• Displacement ($\Delta x$): The change in position of a particle.
  • Defined as the final position minus the initial position: $\Delta x = x - x_o$.
  • Displacement is a vector quantity, meaning it has both magnitude and direction.
  • Its algebraic sign indicates direction: positive for movement in the positive x-direction, negative for movement in the negative x-direction.
  • Examples of Displacement Calculation:
    • Ex. 1: Starting at $x_o = 2.0$ m and ending at $x = 7.0$ m.
      • $\Delta x = x - x_o = 7.0\text{ m} - 2.0\text{ m} = 5.0\text{ m}$.
    • Ex. 2: Starting at $x_o = 7.0$ m and ending at $x = 2.0$ m.
      • $\Delta x = x - x_o = 2.0\text{ m} - 7.0\text{ m} = -5.0\text{ m}$.
    • Ex. 3: Starting at $x_o = -2.0$ m and ending at $x = 5.0$ m.
      • $\Delta x = x - x_o = 5.0\text{ m} - (-2.0\text{ m}) = 7.0\text{ m}$.
• Important Distinction: Displacement ($\Delta x$) should not be confused with Distance Traveled.
  • Distance traveled is the total length of the path taken and is always a positive value.
  • Displacement only depends on the initial and final positions, not on the path taken between them.

Speed and Velocity

Average Velocity ($\bar{v}$)

• Definition: The ratio of the displacement ($\Delta x$) to the time interval ($\Delta t$) during which the displacement occurs.
• Formula: $\bar{v} = \frac{\text{Displacement}}{\text{Elapsed time}} = \frac{\Delta x}{\Delta t} = \frac{x<em>f - x</em>i}{t<em>f - t</em>i}$.
• Dimensions: L/T (Length divided by Time).
• SI Units: meters per second (m/s).
• Characteristics:
  • It is independent of the path taken between the initial and final points; it only depends on these coordinates.
  • It can be positive or negative, depending on the sign of the displacement $\Delta x$. The time interval $\Delta t$ is always positive.
  • If a particle starts at some point and returns to the same point, its displacement is zero ($\Delta x = 0$), and thus, its average velocity for the entire trip is zero.
  • Average velocity gives no details about the motion (e.g., speed variations) between the initial and final points.
  • Graphical Interpretation: On a position-time graph, the average velocity is equal to the slope of the straight line connecting the initial and final points.
• Example from transcript (Truck moving in -x direction):
  • Initial position $x<em>1 = 277$ m at $t</em>1 = 6.0$ s.
  • Final position $x<em>2 = 19$ m at $t</em>2 = 16.0$ s.
  • $\Delta x = x<em>2 - x</em>1 = 19\text{ m} - 277\text{ m} = -258\text{ m}$.
  • $\Delta t = t<em>2 - t</em>1 = 16.0\text{ s} - 6.0\text{ s} = 10.0\text{ s}$.
  • $\bar{v}_{avg,x} = \frac{-258\text{ m}}{10.0\text{ s}} = -26\text{ m/s}$.
• Example from transcript (Jet-Engine Car): ThrustSSC set a world record by achieving high speeds. To nullify wind effects, a driver makes two runs.
  • Run 1: Displacement $+1609$ m in $4.740$ s. Average velocity $\bar{v} = \frac{+1609\text{ m}}{4.740\text{ s}} = 339.5\text{ m/s}$.
  • Run 2: Displacement $-1609$ m in $4.695$ s. Average velocity $\bar{v} = \frac{-1609\text{ m}}{4.695\text{ s}} = -342.7\text{ m/s}$.

Average Speed

• Definition: The ratio of the total distance traveled to the total time it takes to travel that distance.
• Formula: Average speed $= \frac{\text{Total distance traveled}}{\text{Total time}}$.
• Characteristics:
  • Unlike average velocity, average speed has no direction and, therefore, carries no algebraic sign (it is always positive).
  • It tells us nothing about the details (e.g., stops, turns) of the trip itself, only the overall rate of distance coverage.
• Example from transcript (Jogger):
  • A jogger runs for $1.5$ hours ($5400$ s) at an average speed of $2.22$ m/s.
  • Distance $= \text{Average speed} \times \text{Elapsed time} = (2.22\text{ m/s})(5400\text{ s}) = 12000\text{ m}$.

Average Velocity vs. Average Speed Comparison

• Example: Walking $45$ m down a corridor, then back $15$ m, total time $40$ s.
  • Displacement ($\Delta x$): $x<em>f - x</em>i = (45\text{ m} - 15\text{ m}) - 0\text{ m} = 30\text{ m}$.
  • Total Distance Traveled: $45\text{ m} + 15\text{ m} = 60\text{ m}$.
  • Average Velocity: $\frac{30\text{ m}}{40\text{ s}} = 0.75\text{ m/s}$.
  • Average Speed: $\frac{60\text{ m}}{40\text{ s}} = 1.5\text{ m/s}$.

Instantaneous Velocity ($v$)

• Definition: The velocity of a particle at a specific instant in time. It indicates how fast an object is moving and in what direction at that very moment.
• Formula (Calculus definition): $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$.
  • This is the derivative of the position ($x$) with respect to time ($t$).
• Characteristics:
  • It is crucial when velocity is not constant over an interval.
  • Can be positive, negative, or zero, depending on the direction of motion relative to the chosen coordinate system.
  • From this point forward, the term