$A$ spacecraft which is moving with a speed $u$ relative to the earth in the $x$-direction,enters the gravitational field of a much more massive planet which is moving with a speed $3u$ in the negative $x$-direction. The spacecraft exits following the trajectory as shown below. The speed of the spacecraft with respect to the earth a long time after it has escaped the planet's gravity is given by

$A$ person whose mass is $100\, {kg}$ travels from Earth to Mars in a spaceship. Neglect all other objects in the sky and take acceleration due to gravity on the surface of the Earth and Mars as $10\, {m/s^2}$ and $4\, {m/s^2}$ respectively. Identify from the below figures,the curve that fits best for the weight of the passenger as a function of time.

$A$ hemispherical shell of mass $2M$ and radius $6R$ and a point mass $M$ are performing circular motion due to their mutual gravitational interaction. Their positions are shown in the figure at any moment of time during motion. If $r_1$ and $r_2$ are the radii of the circular paths of the hemispherical shell and the point mass respectively,and $\omega_1$ and $\omega_2$ are the angular speeds of the hemispherical shell and the point mass respectively,then choose the correct option.

The dependence of the intensity of the gravitational field $(E)$ of the Earth on the distance $(r)$ from the center of the Earth is correctly represented by:

In the following four periods:
$(i)$ Time of revolution of a satellite just above the earth's surface $({T_{st}})$
$(ii)$ Period of oscillation of mass inside the tunnel bored along the diameter of the earth $({T_{ma}})$
$(iii)$ Period of simple pendulum having a length equal to the earth's radius in a uniform field of $9.8 \; N/kg \; ({T_{sp}})$
$(iv)$ Period of an infinite length simple pendulum in the earth's real gravitational field $({T_{is}})$

$A$ spacecraft travels in a straight line from the Earth to the Moon. How will its weight change during this journey?