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What energy (in electronvolts) will a proton gain by a fall from infinite distance to the surface of Earth?

As you have probably heard, planets and any other bodies in the central gravitational field move on conic sections (in case of the Solar system ellipses with small eccentricity). Find out, how would trajectories look like in a universe, where gravitational force was proportional to multiplicative inverse of distance raised to the third power (instead of second power).

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Imagine a container from which continually and horizontally flows out water stream with constant cross-section area. Velocity of the stream randomly fluctuate with uniform distribution from $v_1$ to $v_2$. Water from the container continually freely falls onto a horizontal floor below. Figure out arbitrary area of the floor to which falls exactly $90 \mathrm{\%}$ of water.

1. Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius $1 \mathrm{ly}$. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically.
2. Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass $m$ in a spherically symmetrical central-force field. \[\begin{equation*} \dot {r}^2 = \frac {2}{m} \(E - V(r) - \frac {l^2}{2mr^2}\) \end {equation*}\] Where $r$ is the length of the radius vector, $E$ is the total energy, $l$ is angular momentum, and $V(r)$ is the potential energy of the mass point.
3. Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.

1. Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane. Instructions:: Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality.
2. Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
3. Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius. Hint:: This problem can be easily solved even without the knowledge of spherical trigonometry.

Your weight would be lower when the Moon is in zenith than when it is in nadir. About how much?

We have two point masses with the same mass $m$ at a distance $d$ from each other. They are located freely in space with no external gravitational forces. What's the minimum velocity we need to impart on one of the points in the direction away from the other point, so that they keep flying away from each other indefinitely?

If we once colonized the Moon, would it be appropriate to use stairs on it? Imagine the descending staircase on the Moon. The height of one stair is $h=15 \mathrm{cm}$ and it's length is $d=25 \mathrm{cm}$. Estimate the number $N$ of stairs that a person would fly over if he walked into the staircase with a velocity $v=5{,}4 \mathrm{km\cdot h^{-1}}=1{,}5 \mathrm{m\cdot s^{-1}}$. The gravitational acceleration on the Moon's surface is six times weaker than on Earth's surface.

We have a satellite and we want to launch it out of the Solar System. We launch it from Earth's orbit so that after some corrections of the trajectory it gets a velocity which is higher than the escape velocity from the Solar System. What is the probability that the satellite will collide with some cosmic material with higher diameter that $d=1 \mathrm{m}$ before leaving the Solar System.