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If a body of mass \( 1 \, \text{kg} \) falls on the earth from infinity, it attains velocity \( (v) \) and kinetic energy \( (k) \) on reaching the surface of earth. The values of \( v \) and \( k \) respectively are _________.
(Take radius of earth to be \( 6400 \, \text{km} \) and \( g = 9.8 \, \text{m} / \text{s}^2 \) )

A planet \( P_{1} \) is moving around the star of mass \( 2M \) in the orbit of radius \( R \). Another planet \( P_{2} \) is moving around another star of mass \( 4M \) in a orbit of radius \( 2R \). Ratio of time periods of revolution of \( P_{2} \) and \( P_{1} \) is ______.

The escape velocity from a spherical planet \( A \) is \( 10 \, \text{km/s} \). The escape velocity from another planet \( B \) whose density and radius are \( 10\% \) of those of planet \( A \), is _______ \( \text{m/s} \).

The height in terms of radius of the earth \( (R) \), at which the acceleration due to gravity becomes \( \frac{g}{9} \), where \( g \) is acceleration due to gravity on earth's surface, is

Three masses \( 200 \,\text{kg} \), \( 300 \,\text{kg} \) and \( 400 \,\text{kg} \) are placed at the vertices of an equilateral triangle with sides \( 20 \,\text{m} \). They are rearranged on the vertices of a bigger triangle of side \( 25 \,\text{m} \) and with the same centre. The work done in this process ________ \( \text{J} \).
\( \left(\text{Gravitational constant } G = 6.7 \times 10^{-11} \, \text{N m}^{2}/\text{kg}^{2}\right) \)

Initially a satellite of \( 100 \, \text{kg} \) is in a circular orbit of radius \( 1.5 R_E \). This satellite can be moved to a circular orbit of radius \( 3 R_E \) by supplying \( \alpha \times 10^{6} \, \text{J} \) of energy. The value of \( \alpha \) is ______________.
\( \big(\text{Take Radius of Earth } R_E = 6 \times 10^{6} \, \text{m} \text{ and } g = 10 \, \text{m}/\text{s}^{2} \big) \)

Net gravitational force at the center of a square is found to be \( F_1 \) when four particles having mass \( M, 2M, 3M \) and \( 4M \) are placed at the four corners of the square as shown in figure and it is \( F_2 \) when the positions of \( 3M \) and \( 4M \) are interchanged. The ratio \( \frac{F_1}{F_2} \) is \( \frac{\alpha}{\sqrt{5}} \). The value of \( \alpha \) is __________.

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JEE Main 2026 4th April Evening Shift

The height in terms of radius of the earth \( (R) \), at which the acceleration due to gravity becomes \( \frac{g}{9} \), where \( g \) is acceleration due to gravity on earth's surface, is