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 $P<em>1\,(x</em>1,t<em>1)$ and $P</em>2\,(x<em>2,t</em>2)$: $\Delta x = x<em>2 - x</em>1$
  • 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): $\bar v = \frac{\Delta x}{\Delta t}=\frac{x<em>2-x</em>1}{t<em>2-t</em>1}$
• Units: $\text{meters}\text{/}\text{second}$ (m$\cdot$s$^{-1}$).
• 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 $P<em>1$ and $P</em>2$ 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 $P<em>1$ and $P</em>2$ 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:
  $\bar a = \frac{\Delta v}{\Delta t}=\frac{v<em>2 - v</em>1}{t<em>2-t</em>1}$
• 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: $v<0$ (moving left), slope $>0$ (\Rightarrow) a>0 acting opposite motion (\Rightarrow) object decelerating (slowing down while moving left).
  • 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.