Find the magnitude of the gravitational force between two-point particles of masses $2.5 \ kg$ and $3.0 \ kg$ which are at a distance of $10 \ cm$ from each other.

Three stars in the Milky Way were found in the following symmetrical arrangement.

What is the net force acting on star J due to the other two stars?

(The mass of each star is equal to $6 × 10^{28} \ kg$ and the $XY$ axis is in standard orientation)

A satellite is moving in a circular orbit at a fixed distance of $1{,}000 \ km$ from the surface of the Earth. What is the time period of this satellite?

$(M_e=6×10^{24} \ kg, \ R_e=6400 \ km)$

Alex drops one object from points $A$ and $B$ each. He then finds the ratio of acceleration due to gravity for those objects.

Which of the below options shows the ratio obtained by Alex?

A satellite is launched from the surface of a planet such that it escapes the gravitational field and travels with a speed of $10 \ km/s$ thereafter. Determine the velocity with which the satellite must be launched. (Take the mass of the planet as $3×10^{26} \ kg$ and its radius as $8×10^7 \ m$.)

The figure below shows the orbit of a planet around the Sun, where $A$, $B$, $C$ and $D$ show the positions of the planet at different times.

In which position does the planet travel with the fastest speed around the Sun?

What is the total mechanical energy of a satellite that is orbiting a planet at a distance of $1.5 × 10^{10} \ m$ from the center of the planet?

(Assume the mass of the planet to be $5×10^{25} \ kg$ and the mass of the satellite to be $1.2 × 10^2 \ kg$.)

The time period of two satellites, $LI$ and $MBII$, moving around the Earth is $24 \ hours$ and $2.5 \ hours$, respectively. The distance of $LI$ is approximately $36{,}000 \ km$ from the center of the Earth. What is the distance of $MBII$ from the center of the Earth?

The acceleration due to gravity varies at large distances from the surface of the Earth. If an object is dropped at a distance of $h$ from the surface of the Earth, then what is the velocity of the object $v$ as a function of its displacement $x$?

(Hint: You might find the relation $ \frac{dv}{dt} = v \frac{dv}{dx}$ helpful.)

The distance between planet $A$ and planet $B$ is always equal to $50R$ while they move in their respective orbits. A small rover is to be launched from planet $A$ to planet $B$.

What should be the speed $v$ of the rover with which it should be launched from planet $A$ so that it reaches planet $B$ with a speed of $u = \sqrt{\frac{4Gm}{R}} \ ?$

(Note: The values inside the circles shows the mass and the radius of the planet respectively.)

Which of the below statements is TRUE regarding Kepler’s second law?

Look at the figure below.

What should be the perihelion speed of planet $A$?