# 3. Series 31. Year

### (3 points)1. slowed down

Let's suppose a camera with a frame rate of 24 frames per second (consider evenly spaced and perfectly sharp shots). We record a flight of a helicopter with the rotor rotation velocity of $2 900 \mathrm {cycles/min}$. Then the record is played. What is the apparent rotational velocity of the rotor in the record?

### (3 points)2. small acceleration, large acceleration

In the figure, there is an ellipse with two focal points $F_1$ $F_2$ and several marked points on the ellipse. The ellipse represents a trajectory of one material point. Plot the accelerations the point experiences in given points of its trajectory. Show it in a figure. The direction and ratio of accelerations are important.

1. There is a massive body in the focal point $F_1$. The material point is orbiting it, and Kepler's $2^{\rm nd}$ law applies.
2. The absolute value of velocity of the material point is constant. It only moves along the ellipse.

### (6 points)3. IDKFA

You fired at Imp from your plasma gun which shoots a cluster of particles with uniform velocity distribution in interval $\langle v_0, \; v_0+\delta v\rangle$ (all of the particles are moving in one line, there is no transverse velocity). The total kinetic energy of the cluster is $E_0$. The barrel rifle has a cross cestion of $S$ and the pulse takes an infinitely short time. How far does Imp need to stand to be safe. Assume that his skin is able to cool the heat flow of $q$.

### (7 points)4. dropped pen

We drop a pen (rigid stick) on a table so that it makes an angle $\alpha$ with horizontal plane during its fall. Calculate the velocity of the higher end during its impact. When we dropped the pen, its center of mass was at height $h$. All collisions are inelastic and friction between the table and the end of the pen large enough.

Bonus: Calculate the angle $\alpha$ so that the velocity (of the second end that touches the table) is maximal. For which height $h$ is it worth to tilt the pen?

Matt was bored.

### (8 points)5. decay here, decay there

We have $A_0$ particles which decay into $B$ particles with decay constant $\lambda \_A$. $B$ particles decay into $A$ particles with decay constant $\lambda \_B$. The number of $B$ particles at the beginning is $B_0$. Find a ratio of the numbers of particles $A$ and $B$ as a function of $t$.

### (8 points)P. folded paper

Everyone has certainly heard and surely tried it: „Sheet of paper can not be folded in a half more than seven times.“ Is it really true? Find boundary conditions.

Kuba was bored and folded a paper.

### (12 points)E. magnetically attractive

You got a planar magnet (magnetic foil) together with the tasks of these series. This magnet is a bit different than a rod magnet. The south and north poles are alternating parallel lines. When approaching the ferromagnetic surface, a magnetic circuit is created which holds the magnet (for example, on the fridge) and can carry even a picture on itself. Your tasks are:

• Measure the area and thickness of the film which you be used for your experiments.
• Measure the mean distance between the two closest same magnetic poles (twice the distance of opposite poles).
• Measure the maximum payload (ie. weight without magnet weight) which can be carried by a $1 \mathrm{cm^2}$ of a magnet if the magnet load is even if the magnet is attached to the bottom of the horizontal plate. The plate should be approx.  $1 \mathrm{mm}$ thick sheet made of magnetically soft steel.

Charles obtained a magnetic foil.

### (10 points)S. going for a walk with integrals

We are sorry, this task is not yet translated…

Mirek and Lukáš random-walk to school.