3.2 Notes on Instantaneous Velocity and Speed
Instantaneous Velocity
Objective: Understand how to determine velocity at a single point in time, and how it differs from average velocity and from speed.
Key idea: Instantaneous velocity is the limit of the average velocity as the time interval between two events along the path goes to zero.
Position as a function of time: Let the position of an object be a continuous function of time, denoted by x(t).
Average velocity between two times: The average velocity between times t1 and t2 is
v{ ext{avg}} = rac{x(t2) - x(t_1)}{t2 - t_1} .Instantaneous velocity: To get the velocity at a single time, take the limit as the elapsed time igl(t2 - t1igr) = igl( auigr) goes to zero, yielding
v(t) = rac{dx}{dt} = oxed{ rac{dx(t)}{dt} }
which is equivalently the derivative of the position function at time t
eq 0.Explicit limit form:
v(t) =
ext{lim}_{ riangle t o 0} rac{x(t+ riangle t) - x(t)}{ riangle t}.
This is the standard derivative notation: v(t) = rac{dx(t)}{dt}
.Relationship to calculus: Instantaneous velocity is the fundamental rate of change of position with respect to time.
Graphical interpretation (position vs. time): The instantaneous velocity at time t0 is the slope of the tangent line to the position–time curve at t0.
A positive slope means the object is moving in the positive direction; negative slope means motion in the negative direction.
If the position function has a local maximum, the slope there is zero, so v(t) = 0. The same occurs at a local minimum.
The instantaneous velocity at a time is not the same as the average velocity over a finite interval; the instantaneous velocity is the limit of the average velocity as the interval shrinks to zero.
Zeros of velocity and extrema: The zeros of the velocity function correspond to the minimum and maximum of the position function (turning points) where the slope of the x(t) curve is zero.
Velocity is a vector; speed is its magnitude:
Instantaneous velocity has both magnitude and direction (units: length per time).
Instantaneous speed is the magnitude of the instantaneous velocity: ext{speed}(t) = |v(t)| .
Average vs instantaneous speed vs velocity:
Average speed = total distance traveled divided by elapsed time: ext{Average speed} = rac{ ext{Total distance}}{ ext{Elapsed time}} .
Magnitude of average velocity is not necessarily equal to average speed, because the total distance and net displacement can differ (e.g., start and end at same location gives zero average velocity but nonzero average speed).
Instantaneous speed is the magnitude of instantaneous velocity: ext{Instantaneous speed} = |v(t)| .
Typical speeds (illustrative values): (Table-like examples)
Continental drift: ≈ 1.7 ext{ m/s} (≈ 3.9 mph)
Brisk walk: ≈ 3.9 ext{ mph} (≈ 6.3 km/h)
Cyclist: ≈ 4.4 ext{ m/s} (≈ 10 mph)
Sprint runner: ≈ 12.2 ext{ m/s} (≈ 27 mph)
Rural speed limit: ≈ 24.6 ext{ m/s} (≈ 56 mph)
Official land speed record: ≈ 341.1 ext{ m/s} (≈ 763 mph)
Speed of sound at sea level: ≈ 343 ext{ m/s}
Space shuttle reentry: ≈ 7800 ext{ m/s}
Escape velocity of Earth: ≈ 11200 ext{ m/s}
Orbital speed of Earth around the Sun: ≈ 29783 ext{ m/s}
Speed of light in vacuum: ≈ 2.99792458 imes 10^{8} ext{ m/s}
Calculating instantaneous velocity from a position function:
If the position is given by a polynomial in time,
For a term of the form A t^n, the derivative is rac{d}{dt}(A t^n) = n A t^{n-1}.
The derivative of a sum is the sum of the derivatives; thus, for a polynomial
x(t) = igl( ext{constant}igr) imes t^0 + igl( ext{coefficient}igr) imes t^1 + igl( ext{coefficient}igr) imes t^2 + \, ext{etc},
the velocity is
v(t) = rac{dx}{dt} = ext{sum of the derivatives of each term} = ext{sum of } n imes A_n t^{n-1}.
Examples (conceptual summaries):
Example 3.2 (from Graphs): Given a position–time graph, the velocity–time graph is found by evaluating slopes of the position curve over intervals; the instantaneous velocity at a time is the slope of the tangent at that time; if the object stops, the instantaneous velocity at that moment is zero, which corresponds to a tangent slope of zero.
Example 3.3 (Instantaneous vs Average Velocity): If x(t) is a polynomial in t, use the power rule to find the instantaneous velocity v(t) and use the definition of average velocity to find the average velocity between two times. The solution approach involves differentiating x(t) to get v(t) and computing v_avg from x(t2) and x(t1).
Example 3.4 (Instantaneous Velocity vs Speed): For a given x(t), compute v(t) via differentiation, then speed as |v(t)|. The velocity provides direction; the speed provides magnitude only. Graphs of x(t), v(t), and speed vs time illustrate that velocity can be negative (moving in the negative direction) while speed remains positive.
Check-your-understanding style (conceptual): Given a position function x(t), find:
(a) The velocity function v(t) by differentiation.
(b) Whether the velocity is ever positive (sign of v(t)).
(c) The velocity and speed at a given time (evaluate v(t) and |v(t)| at that time).
The exercise emphasizes that velocity is the slope (derivative) of x(t) and speed is the magnitude of that slope.
Connecting to broader principles:
Instantaneous velocity is a fundamental rate of change and is essential for dynamics analyses (e.g., Newton’s laws, kinetic energy, momentum).
The concept of instantaneous velocity relies on the differentiability of the position function; non-smooth motions (sharp corners) lead to undefined instantaneous velocity at those points.
The sign of velocity communicates direction of motion; the magnitude (speed) communicates how fast the object is moving regardless of direction.
Summary takeaways:
Instantaneous velocity is the derivative dx/dt, the slope of the x(t) curve at time t, and the limit of the average velocity as the time interval shrinks to zero.
Velocity is a vector (has direction); speed is its scalar magnitude.
The instantaneous speed is |v(t)|; average speed is total distance divided by elapsed time.
For polynomial x(t), use the power rule to obtain v(t); then analyze sign and magnitude to interpret motion.