Problem Statement of Series 1, Year 37

Some species, such as cetaceans, navigate by echolocation. Let us assume that a cetacean emits a sound signal through a larynx located precisely between the ears at a distance $a$. Consider a submarine is moving at the same depth as the whale. The sound bounces off the submarine and arrives at the closer ear of the whale at time $t$ from the moment of transmission. If the time delay between the sound picked up by the right and the left ear is $\mathrm{\Delta }t$, what is the distance and direction of the submarine?

Jarda is standing at the end of the platform, waiting for his train to arrive. When the train's first carriage passes him, he discovers that this is the carriage where he has his seat ticket. At this point, the speed of the train is $8.5\mathrm{m}\cdot {\mathrm{s}}^{-1}$, and the train begins to slow down steadily until it stops in $28\mathrm{s}$. Jarda immediately starts walking in the direction of his carriage, but because he has to push through the crowds of passengers, his speed is only $1\mathrm{m}\cdot {\mathrm{s}}^{-1}$. What is the shortest time the train must stay in the station for Jarda to board his carriage?

A cyclist with the mass $m_{\mathrm{c}}=62.3\mathrm{kg}$ started riding his bike at constant power from rest to the wanted speed at time $t=103\mathrm{s}$. His bicycle's steel frame and fork have a mass $M=6.50\mathrm{kg}$, and each of the two wheels has a mass $m=1950\mathrm{g}$. How long would it take him to get going on a bike with a carbon frame and fork that is four times lighter? The weight of the other bicycle parts is included in the cyclist's weight.

Legolas had a dream in which the truck braked so quickly that the container lifted off the ground and did a somersault over the cab. He wondered if that was possible, so he tried to do the math. In his model, the entire truck has a mass of $m$ and comprises a tractor and a container. It can rotate freely in all directions around the point where it is connected to the tractor. When the truck is on a flat road, the center of gravity of the container is $h$ above this connection and at a distance $l$ from it. Depending on the slope of the road $\varphi$, how much force must the truck brake in order to lift the wheels under the container off the road?

In the winter, Matěj found a $0.5{\mathrm{m}}^3$ bale of polystyrene and decided to use it. He made a cube-shaped box out of it. Then he cut the ice from a frozen pond, which he stored in the polystyrene cube in the cellar, where the temperature is constant $9{}^{\circ }\mathrm{C}$. How big should Matěj make the cube so that he has the largest amount of ice left in it after half a year? And how many kilograms of ice will he have left? Suppose that the ice from the pond has a temperature of exactly $0{}^{\circ }\mathrm{C}$. Ignore the volume of polystyrene used for the edges of the cube.

Hint: The thermal conductivity coefficient is the easiest parameter of polystyrene to find.

Using current technology, how much fuel would it take to carry an object of mass $m=1\mathrm{kg}$ into low Earth orbit?

Measure the coefficient of static friction between two sheets of office paper.

1. On long-term average, how long does it take for the March equinox to move by one day when using the Gregorian calendar?
2. How much does the period of oscillation of a pendulum with a period of $t=1\mathrm{s}$ change when its temperature changes by $T=10{}^{\circ }\mathrm{C}$ if its rod and a much heavier weight are made out of copper? What processes affect the pendulum when the atmospheric pressure or air humidity changes?
3. Estimate how long is the shortest “rod” from quartz resonating at a frequency $f=5\mathrm{MHz}$. Consider the density of quartz $\rho =2.65\mathrm{g}\cdot {\mathrm{cm}}^{-3}$ and the modulus of elasticity $E\approx 80\mathrm{GPa}$ and the compressive oscillations with one static and the other free to move.
4. Let's have an isotope ${\phantom{A}}_{\phantom{}}^{\phantom{a}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}a}\mathrm{X}$, that changes with a half-life $T_{1/2}$ to the isotope ${\phantom{A}}_{\phantom{}}^{\phantom{b}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}b}\mathrm{Y}$. At several places in a sample, we measure the relative isotopic abundance of the parent and child nuclides relative to a different isotope of the child element: $\left[{\phantom{A}}_{\phantom{}}^{\phantom{a}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}a}\mathrm{X}\right]/\left[{\phantom{A}}_{\phantom{}}^{\phantom{c}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}c}\mathrm{Y}\right]$, $\left[{\phantom{A}}_{\phantom{}}^{\phantom{b}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}b}\mathrm{Y}\right]/\left[{\phantom{A}}_{\phantom{}}^{\phantom{c}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}c}\mathrm{Y}\right]$. We assume that the relative abundance of the child element does not change in time. How do we determine the age $t$ of the sample? Assume that both isotopes of the element Y are stable and present in original sample and disregard other nuclear transformations.