Kvant Physics Problem 30

A car moves along a straight road with its wheels rolling without slipping.

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Verified: yes
Verdicts: PASS + PASS
Solve time: 13m19s
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Problem

In a movie camera and a film projector, 8 frames pass per second. On the screen, a car is moving, with wheels whose actual diameter is 1 m. The images of the wheels make 2 revolutions per second. What is the speed of the car?

G. L. Kotkin

Setup and Assumptions

A car moves along a straight road with its wheels rolling without slipping. The diameter of each wheel is $D=1,\mathrm{m}$, giving a radius $R=\frac{D}{2}=0.5,\mathrm{m}$. A movie camera records the motion at a rate of $f_c=8,\mathrm{frames/s}$, and the film is projected at the same rate, $f_p=8,\mathrm{frames/s}$, so that the time scale of the motion on the screen is identical to that of the recorded motion. On the screen, the wheels appear to rotate with a frequency of $f_{\mathrm{obs}}=2,\mathrm{rev/s}$. The goal is to determine the speed $v$ of the car. The wheel spokes are assumed identical, so that only their orientation modulo a full revolution is visible, and the observed rotation arises from the stroboscopic aliasing inherent to the discrete frame rate.

Physical Principles

For a wheel rolling without slipping, the translational speed $v$ of the car is related to the angular speed $\omega$ of the wheel by $v=\omega R$. If the wheel rotates at $n$ revolutions per second, then $\omega=2\pi n$, so the car speed can be expressed as $v=2\pi R n$. A camera sampling the wheel at discrete intervals $\Delta t=1/f_c$ records wheel positions separated by $n/f_c$ revolutions. Since only the fractional part of a full revolution between frames is visually discernible, the apparent rotation per frame depends on the fractional part of $n/f_c$. Consequently, the observed rotation per frame satisfies the congruence

$\frac{n}{f_c} \equiv \pm \frac{f_{\mathrm{obs}}}{f_c} \pmod 1,$

where the positive sign corresponds to a forward apparent rotation and the negative sign corresponds to a backward apparent rotation. This accounts for the fact that a visual rotation of $f_{\mathrm{obs}}/f_c$ revolutions per frame is indistinguishable from $1 - f_{\mathrm{obs}}/f_c$ revolutions per frame in the opposite direction.

Derivation

The observed rotation per frame is

$\frac{f_{\mathrm{obs}}}{f_c} = \frac{2}{8} = \frac{1}{4}\ \text{revolution per frame}.$

Considering both forward and backward aliasing, the actual wheel rotation per frame satisfies either

$\frac{n}{f_c} \equiv \frac{1}{4} \pmod 1$

or

$\frac{n}{f_c} \equiv -\frac{1}{4} \pmod 1.$

Multiplying both congruences by $f_c=8$ gives two families of admissible wheel frequencies:

$n = 2 + 8k, \qquad n = 6 + 8k,$

where $k$ is any non-negative integer. Explicitly, the first family corresponds to $n=2,10,18,26,\ldots$ rev/s and the second family corresponds to $n=6,14,22,30,\ldots$ rev/s. Each of these values produces the same apparent rotation of $1/4$ revolution per frame on the screen.

The problem statement does not provide additional information to select a unique value of $n$ from these infinitely many possibilities. The wheel could be turning slowly, giving $n=2$ rev/s and a car speed of $v=2\pi R n = \pi,\mathrm{m/s}$, or it could be turning rapidly, producing $n=10$ rev/s, $n=14$ rev/s, or higher, corresponding to car speeds of $v=10\pi,\mathrm{m/s}$, $v=14\pi,\mathrm{m/s}$, and so on. All these values satisfy the observed stroboscopic condition. Therefore the problem, as stated, admits multiple solutions for the car speed, each consistent with the apparent wheel rotation.

For completeness, the car speed corresponding to the first member of the first alias family is

$v = 2 \pi R n = 2 \pi \cdot 0.5 \cdot 2 = 2 \pi,\mathrm{m/s} \approx 6.28,\mathrm{m/s} \approx 23,\mathrm{km/h}.$

Choosing a larger $k$ in either family yields proportionally higher car speeds, for example $n=10$ rev/s gives $v=10\pi,\mathrm{m/s} \approx 113,\mathrm{km/h}$, and $n=14$ rev/s gives $v=14\pi,\mathrm{m/s} \approx 142,\mathrm{km/h}$. Each of these speeds is physically consistent with the observed stroboscopic effect, but the available information is insufficient to uniquely determine $v$.

Conclusion

The speed of the car cannot be uniquely determined from the given data. All values of the form

$v = 2\pi R n, \quad n = 2 + 8k \ \text{or}\ n = 6 + 8k,\quad k = 0,1,2,\dots$

are compatible with an observed wheel rotation of $2,\mathrm{rev/s}$ at a frame rate of $8,\mathrm{frames/s}$. Representative values of the car speed include $v = 2\pi,\mathrm{m/s} \approx 23,\mathrm{km/h}$, $v = 10\pi,\mathrm{m/s} \approx 113,\mathrm{km/h}$, and $v = 14\pi,\mathrm{m/s} \approx 142,\mathrm{km/h}$. Without further information about the actual rotation rate of the wheels or the car speed, no single numerical value can be selected.