Two trains $$A$$ and $$B$$ are moving with speeds $$20 \mathrm{~m} / \mathrm{s}$$ and $$30 \mathrm{~m} / \mathrm{s}$$ respectively in the same direction on the same straight track, with $$B$$ ahead of $$A$$. The engines are at the front ends. The engines of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from $$f_{1}=800 \mathrm{~Hz}$$ to $$f_{2}=1120 \mathrm{~Hz}$$, as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus $$320 \mathrm{~Hz}$$. The speed of sound in still air is $$340 \mathrm{~m} / \mathrm{s}$$.

The spread of frequency as observed by the passengers in train B is

Column I describe some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situation in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the $$4 \times 4$$ matrix given in the ORS.

Column I | Column II | ||
---|---|---|---|

(A) | The object moves on the x-axis under a conservative force in such a way that its "speed" and "position" satisfy $$v = {c_1}\sqrt {{c_2} - {x^2}} $$, where $$c_1$$ and $$c_2$$ are positive constants. | (P) | The object executes a simple harmonic motion. |

(B) | The object moves on the x-axis in such a way that its velocity and its displacement from the origin satisfy $$v=-kx$$, where $$k$$ is a positive constant. | (Q) | The object does not change its direction. |

(C) | The object is attached to one end of a massless spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. | (R) | The kinetic energy of the object keeps on decreasing |

(D) | The object is projected from the earth's surface vertically upwards with a speed $$2\sqrt {GMe/{\mathop{\rm Re}\nolimits} } $$, where, M$$_e$$ is the mass of the earth and R$$_e$$ is the radius of the earth. Neglect forces from objects other than the earth. | (S) | The object can change its direction only once. |