Key Concepts: One-Dimensional Kinematics

  • Displacement vs distance
    • Displacement: change in position, Ax = x2 − x1; includes direction (sign).
    • Distance: total path length traveled (no sign).
    • Example: 70 m east, then 30 m west → distance = 100 m, displacement = 40 m east.
  • One-dimensional motion convention
    • Along x-axis: +x to the right (positive), −x to the left (negative).
    • Displacement is a vector with magnitude and direction.
  • An object’s position, displacement, and average velocity
    • Displacement over an interval: Ax = x2 − x1.
    • Average velocity over a time interval: ar{v} = rac{A x}{A t} = rac{x2 - x1}{t2 - t1}.
    • Average speed vs average velocity:
    • Average speed = distance traveled / time elapsed.
    • Average velocity = displacement / time elapsed (can differ in magnitude and sign).
  • Instantaneous velocity
    • Definition: v = rac{dx}{dt} = ext{lim}_{ ext{Δ}t o 0} rac{Δx}{Δt}.
    • Graphically, velocity is the slope of the x vs. t curve (dx/dt).
    • If velocity is constant, instantaneous velocity equals average velocity.
  • Instantaneous velocity and graph slopes
    • The slope of the tangent to the x vs. t curve at a point gives the instantaneous velocity at that time.
    • For non-constant velocity, the slope (instantaneous) varies with time.
  • Acceleration (general concept)
    • Definition: ar{a} = rac{Δv}{Δt} over a time interval.
    • Acceleration is a vector; for one-dimensional motion use sign to indicate direction.
    • Physical interpretation: acceleration is the rate of change of velocity; velocity is the rate of change of position.
    • Deceleration: if velocity decreases in the chosen positive direction, acceleration is negative; deceleration depends on directions of velocity and acceleration, not just the word.
  • Instantaneous acceleration
    • Definition: a = rac{dv}{dt} = ext{lim}_{Δt o 0} rac{Δv}{Δt}.
    • Graphically, the slope of the v vs. t curve is the instantaneous acceleration.
  • Constant acceleration (kinematic equations)
    • Set initial time t0 = 0; initial position x0 and initial velocity v0.
    • Four useful equations (constant a):
    • v = v_0 + a t
    • x = x0 + v0 t + frac{1}{2} a t^2
    • v^2 = v0^2 + 2 a (x - x0)
    • x = x0 + frac{v0 + v}{2} \, t
  • Example highlights (brief)
    • Runway design: with acceleration a = 2.00 m/s^2 and x = 150 m, v0 = 0, max speed 27.8 m/s:
    • v = \sqrt{v0^2 + 2 a (x - x0)} = \sqrt{0 + 2(2.00)(150)} = \sqrt{600} ≈ 24.5\ \text{m/s}.
    • Required x to reach 27.8 m/s: \Delta x = \frac{v^2 - v_0^2}{2a} = \frac{(27.8)^2}{2(2)} ≈ 193\ \text{m} \Rightarrow \text{≈ 200 m}.
    • Air-bag design (rough estimate): (v_i = 28\ \text{m/s}); stopping distance ~1 m; a ≈ -\dfrac{Δv}{Δt} ≈ -390\ \text{m/s}^2; stop time ≈ 0.07 s.
  • Practical note on using graphs
    • x vs t: slope = v; v vs t: slope = a; a vs t: constant if a is constant.
    • If a is constant, instantaneous and average accelerations are equal over any interval.
  • Example: x(t) = A t^2 + B
    • v(t) = dx/dt = 2 A t
    • a(t) = dv/dt = 2 A (constant)
    • Demonstrates v increases linearly with t and a is constant.
  • Quick calculus references (preview)
    • Polynomial derivative: rac{d}{dt}(C t^n) = n C t^{n-1} for constant C.
  • Important cautions
    • Do not confuse average acceleration with instantaneous acceleration; use definitions.
    • When a = 0, velocity may be nonzero (constant velocity); when v = 0, acceleration may be nonzero (starting from rest, braking, etc.).