Series 6, Year 30

Two machine guns, that are able to shoot bullets of mass $m=25g$ and speed $v_{1}=500\;\mathrm{m}\cdot \mathrm{s}^{-1}$ with at 10 rounds per second, are attached to the front of a car. The car accelerates on a flat surface to a speed $v_{2}=80\;\mathrm{km}\cdot h^{-1}$ and then starts firing. How many shots will be fired before the car stops? The car is neutral whilst shooting, the air and tyre resistance can be ignored. The heat losses in the machine guns are also negligible.

From what height would we need to „drop“ an object on a neutron star to make it land with a speed 0,1 $c$ (0,1 of speed of light). Our neutron star is 1.5 times heavier than our Sun and has diameter $d=10\;\mathrm{km}$. Ignore both the atmosphere of the star and its rotation. You can also ignore the correction for special relativity. However, do compare the results for a homogenous gravitational field (with the same strength as is on the star surface) and for a radial gravitational field. Bonus: Do not ignore the special relativity correction.

Superman and Flash decided to race each other. The race takes place in deep space as there is no straight beach long enough on Earth. As Flash is slower, he starts with a length lead $l$ ahead of Superman. At one moment, Flash starts with a constant speed $v_{F}$ comparable with the speed of light. At the moment Superman sees that Flash started, he starts running at a constant speed $v_{S}>v_{F}$. How long will it take Superman to catch up with Flash (from Superman's point of view)? How long will it take from Flash's point of view? Was the starting method fair? Can you devise a more fair method (keeping the length lead $l)?$

Mirek wants to shoot a rat he sees at the dorm. To that end, he made a simple air gun which can be modeled as a tube with constant cross-section $S=15\;\mathrm{mm}$ and length $l=30\;\mathrm{cm}$ closed on one side and open on the other. Mirek plans to place a bullet of mass $m=2g$ into the tube so that the bullet seals to tube exactly and is fixed at a distance $d=3\;\mathrm{cm}$ from the closed end. He that pumps up the closed section to a pressure $p_{0}$ and then releases the bullet. He wants the speed of the bullet to be at least $v=90\;\mathrm{m}\cdot \mathrm{s}^{-1}$ as it exits the tube. What pressure will he need to achieve if the gas is ideal? Discuss the realism of the situation. Assume the bullet is released by a quasi-static adiabatic process where $κ=7⁄5$, as the gas is diatomic. Assume an external atmospheric pressure $p_{a}=10^5Pa$. Neglect losses due to friction, air resistance and gas compression ahead of the bullet.

Consider a homogeneous rod of constant cross-section and length $l$ attached to a freely rotating joint at one end. At the beginning, the rod points straight up and is in a homogeneous gravitational field with acceleration $g$. Due to a slight whiff of wind, the rod starts turning and „falling“ down, but is still held on the joint. Find the acceleration of the end of the rod in time.

A very large piece of ice (let us say with diameter 1 km) is placed near a Sun-like star to a circular orbit. It is placed so close, that the equillibrium temperature of a black body at this distance would be approximately 30 ° C. What will happen with such an asteroid and its orbit? The asteroid is not tidally locked.

Get a wine glass, ideally a thin one with a ground edge. First measure the internal diameter of the glass as a function of height from the bottom. Then use it to create sound by moving a wet finger along its edge (this requires pations). Measure how do the frequencies of tones created in this way depend on the height of water in the glass (measure at least 5 different heights and 2 frequencies at each height). Hint: If the walls of the glass are thin, we can assume the internal dimensions are the same as external and measure the diameter as a function of height from an appropriate photograph with a scale. For measuring sound we recommend the free software Audacity (Analyze $→$ Plot spectrum).

1. Try to describe in your own words how and for what purpose nonlinear regression is used (it is sufficient to briefly describe the following: model of nonlinear regression, methods for finding regression coefficients, uncertainties in the determination of regression coefficients, uncertainties in the function being fitted, statistic methods for testing the values of the regression coefficients, how to choose the form of the fitting function). It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
2. In the attached data file regrese1.csv you may find pairs of valuest $(x_i, y_i)$. Fit these data with a sine function in the form \[\begin{equation*} f(x) = a+ b \cdot \sin (c x + d) . \end {equation*}\] Plot the measured values and the fit and comment on it briefly. It’s not necessary to perform regression diagnostics.
   Hint: Be wary of correct constraints for the values of parameter $c$.
   
3. In the attached data file regrese2.csv you may find pairs of values $(x_i, y_i)$. Fit these data with an exponential function in the form \[\begin{equation*} f(x) = a+ \eu ^{b x + c} . \end {equation*}\] Estimate the values of all regression coefficient including their uncertainties.
   Hint: Using graphical method examine homoscedasticity. You may use Huber-White (sandwich) estimator for determining the uncertainties in estimating regression coefficients if necessary.
   
4. In the attached data file regrese3.csv find the pairs of values $(x_i, y_i)$. Fit these data with a hyperbolic function in the form \[\begin{equation*} f(x) = a+ \frac {1}{b x + c} . \end {equation*}\] Plot the measured data in the form of means and error bars and briefly comment on it. Perform the regression diagnostics.

Bonus: In the attached data file regrese4.csv you may find pairs of values $(x_i, y_i)$. We want to fit these data with a function too complex to be expressed analytically. Use spline regression to fit these data with appropriately chosen knots and order).

For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

• script in the UTF-8 coding
• script in the CP-1250 coding
• regrese1.csv
• regrese2.csv
• regrese3.csv
• regrese4.csv
• ukazka1.csv
• ukazka2.csv
• ukazka3.csv