# Series 1, Year 37

### (3 points)1. Moby Dick

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 $\Delta t$, what is the distance and direction of the submarine?

### (3 points)2. train shifting

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 $8{,}5 \mathrm{m\cdot s^{-1}}$, 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 $1 \mathrm{m\cdot s^{-1}}$. What is the shortest time the train must stay in the station for Jarda to board his carriage?

### (5 points)3. new bicycle

A cyclist with the mass $m_c=62{,}3 \mathrm{kg}$ 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.

### (7 points)4. truck flip

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 $\phi$, how much force must the truck brake in order to lift the wheels under the container off the road?

### (10 points)5. cold water immersion in the summer

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 $9{} \mathrm{\C }$. 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 $0{} \mathrm{\C }$. 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.

### (10 points)P. rocket

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

### (12 points)E. wipe the paper

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

### S.

We are sorry. This type of task is not translated to English. 