Case Study 2 — The Speed at an Instant: How a Limit Tames a Radar Gun

Field: Physics / kinematics Calculus used: the intuitive limit (Section 3.2), the $0/0$ form resolved by algebra (Section 3.3), one-sided agreement (Section 3.4), the limit as the foundation of the derivative (Section 3.11)

A Question That Should Be Easy

A police radar gun on a highway claims to read a car's speed "right now" — at a single instant, say 2:14:07.000 PM. Pause on how strange that claim is. Speed is distance divided by time: $\text{speed} = \Delta s / \Delta t$. But an instant has no duration. Over the instant 2:14:07.000, the car travels zero distance in zero time, and the honest answer is $0/0$ — the very indeterminate form Section 3.3 warned us carries no value at all. Yet the radar gun displays $63$ mph with a straight face, and it is right. How can a meaningful, single number come out of $0/0$?

This is not a gadget trick; it is the oldest problem in calculus, and the limit is its only honest resolution. Galileo measured falling bodies in the 1600s and could compute average speeds over measurable intervals, but the idea of speed at an instant eluded rigorous definition until Newton and Leibniz — and was not made fully airtight until the limit was, by Weierstrass, two centuries later. This case study works the problem from scratch for a concrete falling body, watches the $0/0$ dissolve under algebra, and shows that the radar gun is, quite literally, a limit machine.

A Concrete Falling Body

Drop a steel ball from a tall tower. Ignoring air resistance, its distance fallen after $t$ seconds is the classic kinematic law

$$s(t) = \tfrac12 g t^2 = 4.9\,t^2 \quad \text{(meters, with } g = 9.8\ \text{m/s}^2\text{)}.$$

We want the ball's instantaneous speed at exactly $t = 2$ seconds. The only speed we can actually measure is an average speed over a time interval of nonzero length. So we measure over the interval from $t = 2$ to $t = 2 + h$, where $h$ is a small elapsed time:

$$\bar v(h) = \frac{s(2 + h) - s(2)}{(2 + h) - 2} = \frac{s(2+h) - s(2)}{h}.$$

This is a difference quotient — geometrically, the slope of the secant line joining two points on the graph of $s$, and physically, the average speed over the interval of length $h$. Let us gather numerical evidence the way Section 3.2 taught, shrinking $h$ toward $0$:

From above the averages descend toward something near $19.6$ m/s. Approaching from before $t = 2$ (using $h < 0$, an interval ending at $t = 2$) gives the same target from the other side, the left-and-right agreement of Section 3.4. The numbers are persuasive — but, exactly as in Section 3.2, a table never proves a limit. The algebra must close the case.

Watching the 0/0 Dissolve

Substitute and simplify the difference quotient symbolically, leaving $h$ as a live variable:

$$\bar v(h) = \frac{4.9(2 + h)^2 - 4.9(2)^2}{h} = \frac{4.9\big[(4 + 4h + h^2) - 4\big]}{h} = \frac{4.9\,(4h + h^2)}{h}.$$

Here is the decisive move. For every $h \neq 0$ we may cancel the common factor $h$ — and we are allowed to, because the limit (Section 3.2) only ever cares about $h$ values near $0$, never $h = 0$ itself:

$$\bar v(h) = \frac{4.9\,h\,(4 + h)}{h} = 4.9\,(4 + h) = 19.6 + 4.9\,h \quad (h \neq 0).$$

The troublesome $0/0$ is gone. What remains, $19.6 + 4.9h$, is a tame linear expression we can evaluate by substitution. Now take the limit:

$$v(2) = \lim_{h \to 0} \bar v(h) = \lim_{h\to 0} (19.6 + 4.9\,h) = 19.6 \ \text{m/s}.$$

That $19.6$ is the instantaneous speed at $t = 2$ s — and notice it matches the table's drift exactly, while being exact where the table was only suggestive. The radar gun's $0/0$ had a definite value all along; the limit is what extracts it.

Why the Cancellation Is Legitimate (and the Whole Point)

A careful student should feel a flicker of unease. We divided by $h$, then set $h$ to $0$. Isn't that the forbidden "divide by zero" dressed up? No — and seeing why not is the entire moral of this chapter. We never divided by zero. We divided by $h$ while $h$ was a nonzero number near $0$, which is always legal. The function $\bar v(h) = (4.9)(4h + h^2)/h$ and the function $19.6 + 4.9h$ are identical for every $h \neq 0$ — they differ only at the single punched-out point $h = 0$, exactly the hole picture from Section 3.2. The limit reads off the height the graph approaches at that hole ($19.6$), not the value at it (which is the meaningless $0/0$). Bishop Berkeley once mocked this maneuver as juggling "the ghosts of departed quantities"; the limit concept of Section 3.10 is precisely what exorcises the ghost, by speaking only of values near $h=0$ and never at it.

The General Pattern, and the Birth of the Derivative

Nothing about $t = 2$ was special. Run the same algebra at a general time $t$:

$$v(t) = \lim_{h\to 0}\frac{4.9(t+h)^2 - 4.9 t^2}{h} = \lim_{h\to 0}\frac{4.9(2th + h^2)}{h} = \lim_{h\to 0}\big(9.8\,t + 4.9\,h\big) = 9.8\,t.$$

So the instantaneous speed of the falling ball is $v(t) = 9.8\,t$ — speed growing linearly with time, the familiar "$v = gt$" of first-year physics, here derived rather than asserted. The pattern

$$v(t) = \lim_{h\to 0}\frac{s(t+h) - s(t)}{h}$$

is the limit definition of the derivative, $v(t) = s'(t)$, which Chapters 5 and 6 will develop in full. Every instantaneous rate of change in science — velocity, acceleration, reaction rate, marginal cost, electric current — is a limit of an average over a shrinking interval, computed by exactly the cancellation we just performed. The radar gun does not measure speed at an instant by magic; it measures position at two very close times, forms the difference quotient, and trusts the limit. It is a $0/0$ machine that works because the limit makes $0/0$ behave.

Numerical Confirmation

import sympy as sp

t, h = sp.symbols('t h', positive=True)
s = 4.9 * t**2
dq = (s.subs(t, 2 + h) - s.subs(t, 2)) / h
print(sp.simplify(dq))                 # 4.9*h + 19.6
print(sp.limit(dq, h, 0))              # 19.6  (instantaneous speed at t = 2)

# General instantaneous speed v(t):
dq_general = (s.subs(t, t + h) - s) / h
print(sp.limit(dq_general, h, 0))      # 9.8*t

The symbolic engine confirms both the specific speed $19.6$ m/s and the general law $v(t) = 9.8t$, certifying the hand algebra — the hand-builds-understanding, machine-builds-confidence theme in action.

Discussion Questions

1. Explain to a skeptical friend why "speed at an instant" is not nonsense, even though an instant has zero duration. Where does the $0/0$ come from, and what makes it resolvable?
2. We canceled the factor $h$ and then set $h = 0$. Justify each step. At which step would a genuine division by zero have occurred, and why didn't it?
3. The table of average speeds approached $19.6$ from above (using $h > 0$). What would the averages look like using intervals ending at $t = 2$ (i.e., $h < 0$), and why must both one-sided results agree for the instantaneous speed to be well defined (Section 3.4)?
4. A radar gun samples position at two times about $10^{-2}$ s apart, not at a true instant. Using $v(t) = 9.8t$, estimate the error in reading the falling ball's speed at $t = 2$ from a $10^{-2}$ s interval. Connect this to the finite-$h$ trade-off explored in Case Study 1.
5. Acceleration is the instantaneous rate of change of velocity. Set up (don't fully evaluate) the limit that gives the ball's acceleration at $t = 2$ from $v(t) = 9.8t$. What number do you expect, and why?

Annotated Further Reading

• Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. §2.1 ("The Tangent and Velocity Problems"). Develops the velocity-as-a-limit idea with the same difference-quotient algebra used here; the natural next reading.
• Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Ch. 5. For the rigorous student: the same instantaneous-rate idea built on the ε-δ limit of Section 3.10.
• Strogatz, S. (2019). Infinite Powers. Houghton Mifflin Harcourt. Chs. 1–2. A popular, vivid account of how Galileo, Newton, and Leibniz fought toward instantaneous speed before limits made it rigorous.
• Galileo Galilei (1638). Two New Sciences. The original "Third Day" study of falling bodies and the $s \propto t^2$ law derived above — calculus a half-century before calculus.

A radar gun is a limit made of electronics. It measures two nearby positions, forms a $0/0$ that ought to be meaningless, and reads off the single number the difference quotient approaches. Every instantaneous rate in all of science is computed the same way. The limit is what turns "right now" from a paradox into a measurement.