8/28 Lecture Notes: Constant Acceleration and One-Dimensional Motion (Transcript-Based Summary)

Concepts and Definitions

  • Constant acceleration: a is constant over the interval of motion being modeled. It is the rate of change of velocity with respect to time, i.e.
    • a = \frac{\Delta v}{\Delta t} = \frac{vf - vi}{tf - ti}
  • Velocity vs. speed: velocity is a vector with direction; speed is magnitude. Gravity has a constant downward acceleration near Earth's surface, approximately g \approx 9.8\ \text{m/s}^2, i.e., the acceleration due to gravity is about 9.8 m/s per second downward each second.
  • Jerk: the time derivative of acceleration (the rate at which acceleration changes). The session notes that jerk exists and that the current focus is on constant acceleration.
  • Stopping point for model: this course keeps the model simple by stopping at constant acceleration, which means velocity changes at a fixed rate.
  • Coordinate conventions and axes: direction conventions (positive vs negative) will be established for 1D or 3D motion; often: up/right are positive, down/left are negative.
  • Gravitational direction convention shown: acceleration due to gravity is downward (negative in the common convention).

Quick Calculation: Two Cars Accelerating from 0 to 60 mph

  • Problem setup: two cars accelerate from 0 to 60 mph in different times.
    • Car A: time = 3.6 s
    • Car B: time = 9.0 s
  • Acceleration (in mph per second):

    • a_A = \frac{60\ \text{mph}}{3.6\ \text{s}} \approx 16.7\ \text{mph/s} \approx 17\ \text{mph/s (2 sig figs)}

    • a_B = \frac{60\ \text{mph}}{9.0\ \text{s}} \approx 6.7\ \text{mph/s}
  • Interpretation: a faster-car (e.g., Corvette or sports car) accelerates more quickly; a slower, gentler acceleration corresponds to a slower car.
  • Unit conversion to SI: 1 mph ≈ 0.447 m/s, so to convert mph/s to m/s^2 multiply by 0.447.
    • For Car A:
      a_A \approx 17\ \text{mph/s} \times 0.447\ \frac{\text{m}}{\text{s}} = 7.6\ \text{m/s}^2
    • For Car B:
      a_B \approx 6.7\ \text{mph/s} \times 0.447\ \frac{\text{m}}{\text{s}} = 3.0\ \text{m/s}^2
  • Emphasis for assessment: show your work and the unit conversions; a numeric answer alone may be insufficient for full credit.

Equation of Motion in One Dimension (x-direction)

  • Velocity as a function of time with constant acceleration:

    • vx(t) = v{x0} + a_x t
  • Position as a function of time with constant acceleration:

    • x(t) = x0 + v{x0} t + \frac{1}{2} a_x t^2
  • Time-independent (velocity–position) relation (derived by eliminating t):

    • v^2 = v0^2 + 2 a \bigl(x - x0\bigr)
  • Notation and interpretation:
    • vx is the final velocity in the x-direction; v{x0} is the initial velocity in x; ax is the acceleration in x; t is time; x0 is initial position; x is position at time t.
    • If a = 0, then vf = vi and x(t) = x0 + v{x0} t (straight-line motion with constant velocity).
  • Direction and signs:
    • The signs of velocity and acceleration determine speeding up or slowing down:
    • If velocity and acceleration have the same direction, speed increases.
    • If velocity and acceleration point in opposite directions, the object slows down.
  • Practical orientation: you can apply these equations to any single dimension (x, y, or z) by consistently using the same axis for that direction; do not mix axes in a single equation.

Directional Conventions and Scenario Sketches

  • Typical convention used in class: positive direction is up or to the right; negative is down or to the left.
  • Example scenarios:
    • Scenario 1: Positive velocity with acceleration in the same direction -> speeding up.
    • Scenario 2: Initial velocity to the left (negative) with acceleration to the right (positive) -> slowing down if velocity and acceleration are opposite; speeds up if they align later.
  • Marker throw example to illustrate velocity vs acceleration directions:
    • Throwing a marker upward: initial velocity is upward (positive); acceleration due to gravity is downward (negative). The marker slows as it rises; at the peak, velocity is zero; after the peak, velocity is downward (negative) and acceleration is still downward (negative), so it speeds up downward.
  • General rule summarized:
    • If velocity and acceleration point in the same direction, the magnitude of velocity increases (speeding up in that direction).
    • If velocity and acceleration point in opposite directions, the magnitude of velocity decreases (slowing down).
  • Note on coordinate choice:
    • It’s common to keep axes fixed (e.g., x-horizontal, y-vertical) and assign positive/negative accordingly; you can re-label directions if needed, but do not mix components (e.g., do not apply a y-acceleration to affect x-velocity).
  • Projectile motion quick note (when extending to two dimensions):
    • In the absence of air resistance:
    • Acceleration in x, a_x = 0
    • Acceleration in y, a_y = -g (downward)
    • Then solve separate equations for x and y with their respective initial conditions.

Solving One-Dimensional Motion Problems: Strategy and Setups

  • Core approach: solve for v_x and x using the three equations of motion; choose the appropriate set based on what is given.
  • Key practices emphasized:
    • Always write down the equation before plugging numbers.
    • Define and assign variables clearly (v{x0}, x0, a_x, t, etc.).
    • Maintain consistency of signs with the chosen coordinate system.
    • Show the steps and units; partial credit is given for showing the reasoning, not just the final number.
    • In assessments, write as if solving for an actual problem (this trains you to follow proper setup and reduces sign errors).
  • Common pitfalls:
    • Mixing x and y components in a single equation.
    • Sign mistakes in velocity/acceleration terms leading to incorrect final answers.
  • Practical tip for problem-solving:
    • If you start at rest, v_0 = 0; this simplifies the equations (some terms drop out).
    • If you know final velocity and acceleration but not time, use the time-independent equation (the v^2 relation) to connect to position as needed.

Projectile Motion and Directional Separation (Conceptual Note)

  • To handle motion at an angle, decompose the velocity into components:
    • v{0x} = v0 \cos\theta
    • v{0y} = v0 \sin\theta
  • Treat x and y motions separately with corresponding accelerations:
    • In the idealized case (no air resistance):
    • a_x = 0
    • a_y = -g
  • Use the same three equations for each axis with their initial conditions, then combine the results to obtain the full trajectory.

Time-Independent Motion Equation and Its Uses

  • The time-independent relation is especially useful when time is not given.
  • It relates velocity and position directly, removing time:
    • v^2 = v0^2 + 2 a (x - x0)
  • This equation is often used to connect speed with displacement when acceleration is constant and time is not specified.

Conceptual Takeaways for Assessments and Practice

  • Always articulate the governing equation(s) and the meaning of each symbol.
  • Clearly state initial conditions: v0, x0, t0 (often t0 = 0), and the acceleration a.
  • Maintain consistent units throughout; convert where necessary before plugging values into equations.
  • Do not rely on a single numeric answer; show the steps, including unit analysis and sign conventions.
  • Be mindful of the direction conventions when interpreting results; flip signs if the problem’s coordinate system differs from your own.

Excel Zone: Tutoring Resource (Community Information)

  • What is the Excel Zone?
    • CCD's free tutoring service offering math and science tutoring, writing tutoring, and digital literacy tutoring.
    • Physics support available through the math and science tutoring program.
  • How to access:
    • Location: Fourth Floor of the Confluence Building.
    • Sign in by tapping your card or entering your S-number.
    • Sit in the area where you want help; a tutor will come when available.
  • Hours (Fall):
    • In-person tutoring: 10:00–18:00, Monday–Thursday; 10:00–15:00 on Friday.
    • In-person tutoring is drop-in (no appointment needed).
    • Online tutoring available by appointment (instructions on the flyer or via CCD Excel Zone website).
  • What they can help with:
    • Math and science tutoring for physics I and II, chemistry I and II, etc.
    • Writing tutoring for essays, job applications, resumes, lab reports, etc.
    • Digital literacy tutoring (email, online submissions, MFA issues, etc.).
  • Accessibility and questions:
    • If hours or flyer details seem confusing, the updated flyer or the math and science tutoring site can help clarify.
  • Final note:
    • The presenter emphasizes using Excel Zone to reinforce understanding and to provide a space to work through assignments with guidance.

Quick Q&A Clarifications (From Transcript)

  • If you start at rest, certain variables simplify (e.g., v_0 = 0 makes some terms vanish).
  • In projectile motion descriptions, x-acceleration is zero (ignoring air resistance); y-acceleration is -g.
  • Do not mix x and y components in a single equation; treat each axis separately when solving.
  • In assessments, show the derivation and set up as you would in an assignment to maximize understanding and partial credit potential.