Problem Statement of Series 3, Year 34

While baking a gingerbread, baking soda, or more rigidly sodium bicarbonate ($\ce{NaHCO3}$), has to be added into the batter. Let's assume, that at high temperatures sodium bicarbonate decomposes as follows \begin{equation*} \ce{2 NaHCO3 -> Na2CO3 + H2O + CO2}\,, \end{equation*} that is, into sodium carbonate, carbon dioxide and water. How much will the volume of the gingerbread increase as a consequence of creation of water steam and carbon dioxide bubbles in the batter after adding $10\,\mathrm{g}$ of sodium bicarbonate? Assume that the water steam and carbon dioxide behave as ideal gases and that the batter solidifies around the bubbles at temperature $200\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ and pressure $1~013\,\mathrm{hPa}$.

Jirka and Káťa want to try bungee-jumping. To jump from a height of $h = 100\,\mathrm{m}$ they have ideally elastic rope with a length of $l=40\,\mathrm{m}$, which is calibrated so that when Káťa with a weight of $m_{\mathrm{K}}=50\,\mathrm{kg}$ jumps with it, she will stop at the height of $h_{\mathrm{K}}=16\,\mathrm{m}$ above the ground. Is this rope safe to use for Jirka if he weights $m_{\mathrm{J}}=80\,\mathrm{kg}$? Neglect the air resistance and the heights of Káťa and Jirka.

Imagine placing a large number of satellites on the geosynchronous orbit. Coincidentally, a runaway series of collisions occurs and forms a thin spherical layer homogenously scattered with ten million shards with an average size of $x = 10\,\mathrm{cm}$. Assume that the velocity directions of the individual shards are oriented randomly in the plane tangent to the sphere. On average, how much time passes between two collisions?

Little Joe the mouse likes to catapult himself from the edge of a fan propeller by simply releasing his grip at the right time and flying away. When should he do it in order to fly as far as possible? The propeller blade has a length $l$ and rotates with an angular velocity $\omega$, while the plane of rotation is perpendicular to the horizontal plane. The center of rotation is at a height $h$ above the ground.

Two spaceships move towards each other on a straight line. The initial distance between them is $d$. The first one moves with the velocity $v_1$, the second with the velocity $v_2$ (in the same reference frame). The first one can reach the maximal acceleration $a_1$, the second one $a_2$ (both regardless of the direction). Their crews want to exchange some “goods”. In order to do that, the spaceships need to meet – i. e. they must be at the same time at the same place and have the same speed. What is the minimal time for them to reach the meeting? Neglect the relativistic effects.

What if the laws of nature weren't the same throughout the whole universe? What if they somehow changed with location? Let's focus on electromagnetic interaction. What would be the minimal change of the Coulomb's law constant as a function of distance, such that we could observe a deviation? How would we observe it?

You have probably heard at school about the thermal motion of molecules such as diffusion or Brownian motion. Measure the time dependance of the size of a color spot in water and calculate the diffusion constant. Make measurements for several different temperatures and plot the temperature dependance of the diffusion constant in a graph. How could you arrange the experiment so that the temperature would stay constant during the measurement?

Consider a particle with charge $q$ and mass $m$, fixed to a spring with spring constant $k$. The other end of the spring is fixed at a single point. Assume that the particle only moves in a single plane. The whole system exists in a magnetic field of magnitude $B_0$, which is perpendicular to the plane of movement of the particle. We will try to describe possible modes of oscillation of the particle. Start by the determination of equations of motion – do not forget to include the influence of the magnetic field.

Next assume that the particle oscillates in both of the cartesian coordinates of the particle and carry out Fourier substitution – substitute derivatives by factors of $i \omega$, where $\omega$ is the frequency of the oscillations. Solve the resultant set of equations in order to determine the ration of the amplitudes of oscillations in both coordinates and the frequency of oscillations. The solution obtained in this way is quite complicated, and better physical insight can be gained in a simpler case. From now on, assume that the magnetic field is very strong, i.e. $\frac{q^2 B_0^2}{m^2} \gg \frac{k}{m}$. Determine the approximate value(s) of $\omega$ in this case, always up to the first non-zero order. Next, sketch the motion of the particle in the direct (i.e. real) space in this (strong field) case.