2.3 Instantaneous Velocity and Velocity-Time Graphs

Instantaneous Velocity

  • Real moving objects do not have a constant velocity; they speed up and slow down with velocity changing smoothly, not in abrupt jumps.
  • Everyday example: starting from a red light, then increasing speed smoothly from 0 to 60 mph as you merge onto the freeway; similarly, a car speeds up as it enters the highway.
  • A speedometer reads the instantaneous velocity: the rate at which position is changing at that exact instant t.
  • If you momentarily are at 40 mph, that means the position is changing at a rate such that you would cover 40 miles in one hour if that rate continued without changing.
  • Instantaneous velocity vs average velocity:
    • Average velocity is the velocity averaged over a finite time interval (e.g., one second or one minute).
    • From this point on, velocity will be interpreted as instantaneous velocity (the velocity at a single instant).
  • Formal definition (conceptual): the instantaneous velocity v_x(t) is the velocity at a single instant of time t; it is the slope of the position-versus-time curve at that time.
  • Mathematical expression:
    • v_x(t) = rac{dx}{dt}
    • This is the derivative of position with respect to time.
  • Significance: instantaneous velocity tells you how fast and in what direction the position is changing at that precise moment.
  • Graphical intuition:
    • If motion is uniform, the position-versus-time graph is a straight line; the velocity is the constant slope of that line.
    • If motion is nonuniform, the position-versus-time graph is curved; the slope (and thus the velocity) changes with time.
  • Quick interpretation: the slope of the position-versus-time graph at time t is the velocity at that instant. In a speeding-up scenario, later time intervals contribute increasingly larger displacements, reflecting the increasing velocity.

Position vs. Time Graph and Instantaneous Velocity

  • For a car entering a highway, the position-versus-time graph is curved, showing that displacement in equal time intervals grows as the car speeds up.
  • The instantaneous velocity is the slope of the position graph at time t; equivalently, it is the slope of the tangent to the curve at that point.
  • How to estimate the instantaneous slope:
    • Consider a very small time interval around t ≈ t0.
    • Compute the rise over run for that small segment to approximate the slope.
    • Graphically, this is equivalent to finding the slope of the tangent line touching the curve at t0.
  • Example near t = 4.0 s:
    • Using a small window, the slope is v_x ≈
    • v_x(4.0 ext{s}) \approx \frac{\Delta x}{\Delta t} = \frac{3.0\ ext{m}}{0.20\ ext{s}} = 15\ \text{m/s}.
    • This is the instantaneous velocity at that instant.
    • Alternatively, the tangent-line slope method yields the same result (the slope of the tangent to the x(t) curve at t = 4.0 s).
  • Conceptual note: a small magnified region around a time point shows the curved graph behaving like a straight line; the slope of that line (the tangent) gives the instantaneous velocity.

Applications: Analyzing Motion with a Hockey Player (Fig. 2.19)

  • Position-versus-time graph for a hockey player can be analyzed by looking at slopes (instantaneous velocities) at different points along the curve.
  • Points on the graph:
    • Before point A (to the left): the slope is negative, so v_x is negative (moving to the left).
    • At point A: the slope increases; around point B, the slope is steepest, giving the maximum velocity (v_x is maximum at B).
    • Between A and C: the slope is positive after A, then decreases toward C; near point C, the slope returns to zero (rest).
    • At rest: where the tangent is horizontal, v_x = 0 (no motion).
  • Constructing a velocity-versus-time graph from the position graph:
    • The velocity at each instant is the slope of the tangent to the position graph at that time.
    • The velocity graph is built point-by-point by finding tangents to the position curve.
  • Takeaways:
    • The maximum velocity occurs where the position graph has the steepest slope (largest tangential slope).
    • The sign of velocity tells the direction of motion (negative slope means moving in the negative x-direction, positive slope means moving in the positive x-direction).

Area Under the Velocity–Versus–Time Graph: Displacement (Fig. 2.21)

  • Central idea: the displacement over a time interval [ti, tf] is the area under the velocity-vs-time graph during that interval.
  • This holds whether velocity is constant or varying with time.
  • Intuition: velocity tells how fast position is changing; integrating velocity over time accumulates that change to give total displacement.
  • Example: a lion pursuing prey can speed up, then reach a steady speed; the total displacement is the area under the v-t curve over the chase interval.

Towering Example: Car Accelerating from Rest (Fig. 2.22)

  • Velocity-versus-time graph for a car starting from rest and increasing velocity to 12 m/s over 3 s:
    • The graph is a straight line rising from 0 to 12 m/s across 3 s.
  • Displacement during first 3 seconds:
    • The area under the v-t curve is the shaded region (a triangle).
    • Calculation:
    • \Delta x = \text{Area} = \frac{1}{2} \times (\text{base}) \times (\text{height}) = \frac{1}{2} \times (3\ \,\text{s}) \times (12\ \text{m/s}) = 18\ \text{m}.\n- Physical interpretation:
    • The final velocity is 12\ \text{m/s} \approx 25\ \text{mph}.
    • If the car had moved at a constant 12 m/s for 3 s, the distance would be 36\ \text{m}. The actual distance is smaller (18 m) because the car started from rest and accelerated gradually; the velocity was lower than 12 m/s for most of the interval.
  • In general: the area under a velocity–time curve represents displacement, and its numerical value depends on the shape of the curve, not just the final speed.

Key Formulas and Concepts to Remember

  • Instantaneous velocity as slope of position-time curve:
    • v_x(t) = \frac{dx}{dt}
  • Displacement as the area under the velocity-time curve:
    • \Delta x = \int{ti}^{tf} vx(t)\, dt
  • For a linear ramp from 0 to v_f over a time interval (\Delta t) (starting from rest):
    • \Delta x = \frac{1}{2} v_f \Delta t
  • Units sanity checks:
    • Velocity: m/s; Time: s; Displacement: m.
    • Area under v(t) (m/s × s) yields displacement in meters.
  • Conceptual clarifications:
    • Instantaneous velocity is the velocity at a single instant; average velocity is the average over a finite interval.
    • The slope of the position-versus-time graph at time t gives v_x(t); the tangent line at t provides that slope.
    • Areas under velocity-time graphs are a geometric and physical representation of displacement.
  • Real-world interpretation cues:
    • Negative velocity indicates motion in the negative direction.
    • Zero velocity occurs when the motion momentarily stops (tangent is horizontal).
    • When the velocity graph rises (positive v) and then falls, the car speeds up and later slows down, respectively.
  • Quick cross-checks:
    • Compare the displacement inferred from the v-t area to what would be obtained if the velocity were constant at the final value; disparities reveal the effects of acceleration.

Quick Conceptual Connections

  • Foundational principle: velocity is the time derivative of position; position is the integral (accumulation) of velocity over time.
  • This ties calculus concepts directly to kinematics: derivatives give instantaneous rates (slopes), integrals give accumulated quantities (areas under curves).
  • Real-world relevance: designing safe speeds, understanding motion in sports (e.g., skating, chasing), and analyzing vehicle acceleration scenarios in physics and engineering.