Kinematics – Displacement, Velocity, Acceleration & Graph Interpretation

Displacement & Average Velocity
  • We examine motion along a single straight line (1-D kinematics). This simplifies analysis as direction is solely determined by sign (+ or -).
  • Displacement between two events P1\,(x1,t1) and P2\,(x2,t2): \Delta x = x2 - x1
    • Displacement is a vector quantity, representing the change in position and direction, unlike distance which is a scalar representing total path length.
    • Tip: Always pay attention to the initial and final positions for displacement, regardless of the path taken.
  • Average velocity (labelled \bar v because we are not tracking every instant): \bar v = \frac{\Delta x}{\Delta t}=\frac{x2-x1}{t2-t1}
  • Units: \text{meters}\text{/}\text{second} (m\cdots^{-1}).
  • Sign information:
    • \bar v>0 (\rightarrow) motion toward the +x direction.
    • \bar v<0 (\rightarrow) motion toward the –x direction.
Position-Time Graphs: Interpreting the Slope
  • We often plot position vs. time (blue curve in lecture). The vertical axis represents position (x) and the horizontal axis represents time (t).
  • The plotted curve is not the physical path (trajectory) of the object. The object still moves along a straight spatial line; the graph only shows how its coordinate changes with time.
    • Tip: Do not mistake the x-t graph for the actual path travelled. A flat graph means no change in position, not always zero velocity across all dimensions.
  • Connecting P1 and P2 with a straight secant line:
    • Slope of that secant = \frac{\Delta x}{\Delta t}=\bar v (average velocity over the interval).
    • A straight line on an x-t graph indicates constant velocity, while a curved line indicates changing velocity.
  • Direction of slope (\leftrightarrow) sign of velocity:
    • Positive slope (\rightarrow) positive velocity (moving in the +x direction).
    • Negative slope (\rightarrow) negative velocity (moving in the –x direction).
    • Zero slope (\rightarrow) zero velocity (object is momentarily at rest).
Instantaneous Velocity (Derivative Concept)
  • To find the velocity at a specific moment, we bring P1 and P2 closer: \Delta t \to 0 (\Rightarrow) the secant becomes a tangent.
  • Instantaneous velocity:
    v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}
  • Geometrically: slope of the tangent to the x–t curve at that instant.
    • Tip: The instantaneous velocity tells you exactly how fast and in what direction an object is moving at one precise moment in time.
  • Example numbers from lecture (illustrating convergence):
    • \Delta t = 2\,\text{s},\;\Delta x = 140\,\text{m} \Rightarrow \bar v = 70\,\text{m}\cdot\text{s}^{-1}
    • \Delta t = 1\,\text{s},\;\Delta x = 55\,\text{m} \Rightarrow \bar v = 55\,\text{m}\cdot\text{s}^{-1} (closer to the true tangent value).
Acceleration (Rate of Change of Velocity)
  • Acceleration measures how quickly velocity changes. Like velocity, acceleration is a vector quantity, meaning its direction matters.
  • Average acceleration:
    \bar a = \frac{\Delta v}{\Delta t}=\frac{v2 - v1}{t2-t1}
  • Units: \frac{\text{m}\cdot\text{s}^{-1}}{\text{s}}=\text{m}\cdot\text{s}^{-2}.
  • Instantaneous acceleration (second derivative of position):
    a = \lim_{\Delta t \to 0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}=\frac{d^2x}{dt^2}
  • Same geometric idea: slope of the tangent to a velocity–time (v–t) graph.
  • Sign of acceleration relative to velocity:
    • If v and a have the same sign (both positive or both negative), the object is speeding up.
    • If v and a have opposite signs (one positive, one negative), the object is slowing down (decelerating).
    • Tip: Acceleration does not always mean speeding up. Negative acceleration can mean slowing down if velocity is positive, or speeding up if velocity is negative.
Velocity-Time Graph: Qualitative Analysis (Points A (\rightarrow) E)
  • Consider the sketched v–t curve with labeled instants A–E. This graph shows velocity on the vertical axis and time on the horizontal axis.
  • Key observations (signs of v and slope):
    • A: v
    • B: v=0 but slope >0 (\Rightarrow) still a>0; object momentarily at rest, about to reverse and move right (changing direction from left to right).
    • C: v>0, slope =0 (\Rightarrow) a=0; fastest point in +x direction, then acceleration changes sign (constant velocity at this instant).
    • D: v>0 but slope <0 (\Rightarrow) a<0 opposite motion, so speed decreasing (slowing down while moving right).
    • E: v<0, slope <0 (\Rightarrow) a<0; velocity and acceleration aligned, object now speeding up toward –x (speeding up while moving left).
    • Crossing the horizontal axis (v = 0) indicates a change in direction, not necessarily a change in position back to the origin.
    • Tip: A v=0 point on a v-t graph only means the object is momentarily stopped; it doesn't mean it's at the origin or permanently stopped.
    • Area under v–t curve = displacement; integrating a section with equal positive & negative area would bring the object back to an earlier coordinate (i.e., zero net displacement for that interval).
    • Tip: Positive area represents displacement in the positive direction, negative area in the negative direction. The net area gives the total displacement.