Notes on Motion Along a Straight Line

PHYS 110 - Chapter 3: Motion Along a Straight Line

Introduction to Motion

• 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.

Understanding Motion

• 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

• Displacement (∆𝑥):

  • Defined as the change in position of a particle over a given time interval.

  • Formula: $ext{∆𝑥} = 𝑥<em>f - 𝑥</em>i$

  • 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.

Velocity

• 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 vs Speed

• 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.

Understanding Instantaneous Velocity

• 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

• 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: $a<em>a ≡ rac{∆v}{∆t} = rac{v</em>f - v<em>i}{t</em>f - t_i}$

  • 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.

Key Concepts in Motion

• 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:

  • $x = x_0 + ut$

  • $v = rac{v_0 + v}{2}$

  • $v = v_0 + at$

  • $x = x<em>0 + v</em>0 t + rac{1}{2}at^2$

  • $v^2 = v<em>0^2 + 2a(x - x</em>0)$

  • These equations apply only for constant acceleration.

Examples

• 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.

Summary

• 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.