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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.

Two masses of negligible dimensions and mass $m=100g$ are connected by a massless string with rest length $l_{0}=1\;\mathrm{m}$ and spring constant $k=50\;\mathrm{kg}\cdot \mathrm{s}^{-2}$. One of the masses is held fixed and the other rotates around it with frequency $f=2\;\mathrm{Hz}$. The first mass can rotate freely around its axis. At one point the fixed mass is released. Find the minimal separation of the two masses during the resulting motion. Do not consider the effects of gravity and assume the validity of Hook's law.

It is well known fact that to make a train ride as comfortable as possible, when accelerating or braking, the acceleration needs to change as little as possible. It is therefore good practice when a train starts with small, constant change of acceleration. The change of acceleration is called a jerk. Determine how does the equilibrium position of a pendulum (the angle with the vertical $φ)$. Denoting the length of the pendulum $l$, the train starts with a constant jerk $k$ ( $k=Δa⁄Δt$, where $a$ denotes acceleration) and the train is on Earth with acceleration due to gravity $g$. Bonus: Derive the equations of motion and solve them numerically for $φ(0)=0$ and $dφ⁄dt(0)=0$ for various values of $k$.

Petr likes to ride a bike on a flat road with a speed $v=10\;\mathrm{m}\cdot \mathrm{s}^{-1}$, and his smart bike tells him that his average power is $P=100W$. After an accident, his breaks are bent and they now persistently act on a wheel with a friction force $F_{t}=20N$ near the circumference. For how long ($t′)$ he needs to cycle now (with the same speed $v)$, to do the same amount of work as before, in time $t?$

Katka decided to go for a walk with her pet rat. They arrived on a flat meadow and when the rat was at a distance $x_{1}=50\;\mathrm{m}$ from Katka, she threw him a ball with the speed $v_{0}=25\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and an angle of elevation $α_{0}$. In that moment, he started running towards her with the speed $v_{1}=5\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Find a general formula for an angle $φ$ as a function of time, where the angle $φ(t)$ is the angle between the horizontal plane and the line between the rat and the ball. Draw this function into a graph and, based on the graph, determine, whether it's possible for the ball to obscure the Sun for the rat, when the Sun is situated $φ_{0}=50°$ above the horizon in the direction of the running rat. Use the acceleration due to gravity $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$ and for simplicity imagine we are throwing the ball from a zero height.

We are going up and down the same hill with the slope $α$, driving at the same speed $v$ and having the same gear (and therefore the same RPM of the engine), in a car with mass $M$. What is the difference between the power of the engine up the hill (propulsive power) and down the hill (breaking power)?

Neutral particle beams are used in various fusion devices to heat up plasma. In a device like that, ions of deuterium are accelerated to high energy before they are neutralized, keeping almost the initial speed. Particles coming out of the neutralizer of the COMPASS tokamak have energy 40 keV and the current in the beam just before the neutralization is 12 A. What is the force acting on the beam generator? What is its power?

A car is approaching a wall with a trajectory that is perpendicular to the wall. The driver, however, wishes to approach the wall safely. Find the car's speed as a function of time, so that the distance between the car and the wall is, at every moment, the same as the distance the car would travel with its instantaneous speed in $T=2\;\mathrm{s}$.

After we press the break pedal, the car does not start to break immediately. During the time $t_{r}$ the breaking force grows linearly up to the maximum force $F_{m}$. Coefficient of static friction between the tire and a road is $f$. What is the maximum speed of car so that the car does not skid even during emergency breaking?

The notebook of a size of A4 (297 x 210 mm) lies on a desk with an inclination of $α=5°$. The notebook weights $m$, between the desk and the notebook there acts a static friction force with coefficient $f_{0}=0.52$. Then, we hit the desk so it starts to oscillate (in the direction of the inclination of the desk) with a frequency $ν=10\;\mathrm{Hz}$ and an amplitude $A=1\;\mathrm{mm}$.

• Determine by which extra force (perpendicular to the desk) we have to act on the notebook so it does not start to move.
• Determine how long it takes the notebook to fall off the desk if at the beggining its bottom edge (the shorter one) is at the bottom edge of the desk. Dynamic friction coeficient is $f$, consider notebook as a rigid plate.