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INPhO 2026, Теория — Задача 4. A Curve You Can Hear

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Всички материали от INPhO 2026

Indian National Physics Olympiad (INPhO)-2026 · Question Paper · 1 февруари 2026 г. · 10 т.

Внимание: Бележка към темата

Matching original question and official key booklets are retained in full. The separate blank Answer Booklet, Summary Answersheet copies and blank graph paper referred to in the questions are absent from the archive; the supplied completed Kepler/glass answer-sheet illustrations remain in the key. Native question pages7 and9 are blank. Question6 says photographs on the next page although its photos occur on native p10 after blank p9. Printed source errors/conventions are preserved and attributed locally: P2 negative dM followed by positive boxed dUm and changing absolute-work notation; P3 Qa=-W followed by positive released heat, dropped-mass wording and P0/V0 notation; P4 derivative/height signs, c/cs and time-delay description; P5 intermediate phiP versus final phiC, log log and r/logarithm numerical scales. The graph/grid pixels and printed fitted results are source data, not newly calculated measurements.

Question booklet · Instructions and Table of Constants

Indian National Physics Olympiad (INPhO)-2026

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION · Tata Institute of Fundamental Research · V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Question Paper

Date: 01 February 2026 · Time: 09:00–12:00 (3 hours) · Maximum Marks: 75

Roll Number: □□□□ – □□□□ – □□□□

Instructions

This booklet consists of 10 pages and total of 6 questions. Write roll number at the top wherever asked.
Booklet to write the answers is provided separately. Instructions to write the answers are on the Answer Booklet.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Take as many data points as possible for the analysis.
Please submit the Answer Sheet at the end of the examination. You may retain the Question Paper.
Except for writing your roll number, no rough work or scribbling is allowed on the question paper.

Table of Constants

Quantity	Symbol	Value
Speed of light in vacuum	$c$	$3.00\times10^8\ \mathrm{m\cdot s^{-1}}$
Magnitude of electron charge	$e$	$1.60\times10^{-19}\ \mathrm C$
Avogadro’s number	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m\cdot s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}$
	$R$	$0.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}$
Boltzmann constant	$k_B$	$1.3806\times10^{-23}\ \mathrm{J\cdot K^{-1}}$
Permeability constant	$\mu_0$	$4\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}$
One Atmospheric pressure	atm	$1.013\times10^5\ \mathrm{Pa}$

Question Number	1	2	3	4	5	6	Total
Maximum Marks	12	12	11	10	18	12	75

Solution booklet · Instructions, constants and alternate-method note

Indian National Physics Olympiad (INPhO)-2026

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION · Tata Institute of Fundamental Research · V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Question Paper

Date: 01 February 2026 · Time: 09:00–12:00 (3 hours) · Maximum Marks: 75

Roll Number: □□□□ – □□□□ – □□□□

Instructions

This booklet consists of 21 pages and total of 6 questions. Write roll number at the top wherever asked.
Booklet to write the answers is provided separately. Instructions to write the answers are on the Answer Booklet.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Take as many data points as possible for the analysis.
Please submit the Answer Sheet at the end of the examination. You may retain the Question Paper.
Except for writing your roll number, no rough work or scribbling is allowed on the question paper.

Table of Constants

Quantity	Symbol	Value
Speed of light in vacuum	$c$	$3.00\times10^8\ \mathrm{m\cdot s^{-1}}$
Magnitude of electron charge	$e$	$1.60\times10^{-19}\ \mathrm C$
Avogadro’s number	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m\cdot s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}$
	$R$	$0.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}$
Boltzmann constant	$k_B$	$1.3806\times10^{-23}\ \mathrm{J\cdot K^{-1}}$
Permeability constant	$\mu_0$	$4\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}$
One Atmospheric pressure	atm	$1.013\times10^5\ \mathrm{Pa}$

Question Number	1	2	3	4	5	6	Total
Maximum Marks	12	12	11	10	18	12	75

Please note that alternate/equivalent methods and different ways of expressing final solutions may exist. A correct method will be suitably awarded.

Условие

An aeroplane flies along a horizontal path, emitting sound at a constant frequency $f_0$ . A stationary observer with a detector on the ground directly beneath the flight path records the sound frequency as the plane passes overhead, which is plotted below. The speed of sound in the medium is $c_s=340\ \mathrm{m\,s^{-1}}$ .

Native aeroplane above ground observer, flight path and height h.

Complete original frequency-time data graph, every blue data point, axes, grid, Hz/s units and all ticks.

(a) Your task is to measure the speed of the aeroplane, $v$ , from the graph. Express $v$ in terms of quantities measurable from the graph, and define/mark these quantities on the graph reproduced in the Summary Answersheet. Calculate the value of $v$ . [5 т.]

(b) Find the height $h$ of the plane’s flight path. [5 т.]

Отговори

Решение

(a)

Let $\theta$ be the angle between the path of the aeroplane and the line joining the aeroplane and the observer. Using the Doppler formula, the expression for the apparent frequency when the observer is at rest is given by

f'=f_0\frac{c_s}{c_s\mp v\cos\theta},

where $c_s$ is the speed of sound. The negative and positive signs correspond to the source moving towards and away from the observer, respectively.

Native solution geometry: aeroplane velocity va, theta, line to observer O, flight path and h.

When the aeroplane is far away, $\cos\theta\approx1$ . Hence, there will be no significant variation in the frequency heard by the observer. This occurs at $t<10\ \mathrm s$ and $t>28\ \mathrm s$ in the given graph.

A significant change occurs only when $\cos\theta$ varies (from 12–24 s). Let the frequencies shown in the graph be $f_l'$ and $f_r'$ when the aeroplane is at the extreme left and right, respectively, where $\cos\theta\approx1$ . Then,

f'=f_0\frac{c_s}{c_s\mp v},

where the $+$ and $-$ signs give $f_r'$ and $f_l'$ , respectively. The ratio of the two frequencies is

\frac{f_l'}{f_r'}=\frac{c_s+v}{c_s-v}.

Rearranging, we obtain

v=c_s\frac{f_l'-f_r'}{f_l'+f_r'}.

From the graph,

f_l'\approx1220\ \mathrm{Hz},\qquad f_r'\approx760\ \mathrm{Hz},

which gives $\boxed{v\approx79\ \mathrm{m/s}}$ . Accepted range of $v=75$ – $80\ \mathrm{m/s}$ .

(b)

The frequency of the sound emitted by the engine is

f_0=\frac{f_l'(c-v)}{c}\approx937\ \mathrm{Hz}.

From the Doppler formula, this frequency is heard at $\theta=90^{\circ}$ . Let $t$ be the observation time and $t'$ be the source emission time. Time $t'$ will include the time delay due to finite distance between the observer and the aeroplane.

\frac{df'}{dt'}=f_0\frac{cv\sin\theta}{(c-v\cos\theta)^2}\frac{d\theta}{dt}.

At $\theta=90^{\circ}$ , $dt'=dt$ . From the geometry shown below,

\frac{d\theta}{dt}=\frac vh.

Native differential-angle construction including vdt, dtheta, h and observer.

Thus,

\left.\frac{df'}{dt}\right|_{\theta=90^{\circ}}=f_0\frac{cv}{c^2}\frac vh,

\Rightarrow\quad h=\frac{f_0v^2}{c\left.\dfrac{df'}{dt}\right|_{\theta=90^{\circ}}}.

From the graph, the frequency near $t\approx18\ \mathrm s$ is approximately $940\ \mathrm{Hz}$ . A tangent drawn at this point gives the slope.

Complete native frequency-time fit, every point, all ticks/units, whole tangent and construction triangle labelled(14.4,1200)/(20.0,800).

The slope at $\theta=90^{\circ}$ is approximately $-71.4$ , which gives $\boxed{h\approx247\ \mathrm m}$ . Depending on the slope calculated, an accepted range of $h$ is 185–250 m.

Source note: the native key switches from $c_s$ to $c$ in (b), prints the derivative and height formula with the signs shown, then quotes a negative fitted slope and a positive height. It describes $t'$ as source emission time while saying that it includes the delay. These printed sign and time conventions are retained.

Теми и трудност

Оценена трудност: 4/5.

Вълни и звук · Механика · Звук и акустика · Ефект на Доплер · Кинематика

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: INPhO2026-Question.pdf · официални решения: INPhO2026-Solution.pdf