Notes on Motion Along a Straight Line

Kinematics: The study of motion of objects in space and time.
- Focuses on describing motion rather than its causes (which is studied in Dynamics).
- Emphasis on developing the terminology and definitions.
- Connection to calculus as concepts are explored.

Position: The location of a particle (object) with respect to a chosen reference point.
Complete Knowledge of Motion: An object's motion is fully understood if its position is known at every moment in time.

Displacement (∆𝑥):
- Defined as the change in position of a particle over a given time interval.
- Formula: ext{∆𝑥} = 𝑥f - 𝑥i
- Where 𝑥f is the final position and 𝑥i is the initial position.
Vector Quantity: Displacement is a vector, meaning it has both magnitude and direction.
Distance: Not to be confused with displacement; it measures the length of the path traveled regardless of direction.
- Example:
- Running a distance of 𝐿 across the room results in 0 displacement if returning to the starting point:
  - ext{∆𝑥} = 0
  - Distance = 2𝐿.

To understand particle motion, it's useful to consider the velocity, which quantifies how displacement changes over time.
Average Velocity (𝑣̅):
- Defined as the ratio of displacement to the time interval:
- Formula: ext{𝑣̅} ≡ rac{ ext{∆𝑥}}{ ext{∆𝑡}}
- Where 𝑣̅ is measured in m/s (SI units).
Interpretation:
- A negative average velocity ((v̅ < 0)) indicates motion along the -x direction.
- A positive average velocity ((v̅ > 0)) indicates motion along the +x direction.
Example Calculations:
- For motion from A to B:
- ext{𝑣̅} = rac{52 - 30}{10 - 0} = 2.2 ext{ m/s}
- For motion from C to D:
- ext{𝑣̅} = rac{0 - 38}{30 - 20} = -3.8 ext{ m/s}

Velocity (Vector) vs. Speed (Scalar):
- Important distinction: velocity takes direction into account while speed does not.
- Example:
- Running across a room and back yields zero displacement (average velocity = 0) while speed is non-zero.
Instantaneous Velocity:
- The limiting value of average velocity as the time interval approaches zero:
- Formula: v = ext{lim}_{ ext{∆t} o 0} rac{ ext{∆𝑥}}{ ext{∆𝑡}} ≡ rac{dx}{dt}
- Here, rac{dx}{dt} is known as the derivative of x with respect to t, acquired from calculus.

As both ext{∆𝑡} and ext{∆𝑥} approach 0, rac{ ext{∆𝑥}}{ ext{∆𝑡}} approaches 0/0, which does not diverge if the x(t) curve is smooth.
Instantaneous velocity gives the slope of the tangent to the curve at a specific time.
Magnitude of Velocity: To find speed from instantaneous velocity:
- Formula: s = |v|
- Speed remains a scalar quantity.
Distance and Speed: More complex calculations compared to position and velocity, usually requiring integral calculus:
- Average speed: v = ext{lim}_{ ext{∆t} o 0} rac{ ext{∆𝑥}}{ ext{∆𝑡}}
- Average speed calculated over time.

Acceleration (a): The rate of change of velocity over time.
Average Acceleration: Defined as the ratio of change in velocity to the time interval:
- Formula: aa ≡ rac{∆v}{∆t} = rac{vf - vi}{tf - t_i}
- Where vf and vi are the final and initial velocities respectively.
Instantaneous Acceleration:
- Given as the limiting value of average acceleration as the time interval approaches zero:
- Formula: a = ext{lim}_{ ext{∆t} o 0} rac{∆v}{∆t} ≡ rac{dv}{dt}
- Units: [a] = ext{m/s}^2 in SI.

Motion Diagrams: Useful visualizations for understanding motion.
Calculating Instantaneous Velocity and Acceleration Graphically: Provides a practical tool for determining these values.
Kinematic Equations: These equations relate displacement, velocity, acceleration, and time under constant acceleration. Key formulas include:
- x = x_0 + ut
- v = rac{v_0 + v}{2}
- v = v_0 + at
- x = x0 + v0 t + rac{1}{2}at^2
- v^2 = v0^2 + 2a(x - x0)
- These equations apply only for constant acceleration.

Instantaneous Velocity:
- Given a motion along the x-axis with equations, students are asked to calculate instantaneous velocities and average velocities over specified intervals.
Graphical Relationships: Students will graph position, velocity, and acceleration versus time to understand their interrelationships better.

The concepts of displacement, velocity, and acceleration form the foundations of kinematics in one dimension.
Understanding these terms and their applications is crucial for further studies in physics, especially in time-dependent motion contexts.