(i) Moon does not have strong gravity to hold atmospheric gases.
(ii) Acceleration due to gravity g is much less on the moon surface than that of the earth's surface.
Hence,\(h=\cfrac { \upsilon ^{ 2 }-{ u }^{ 2 } }{ 2g } \) is larger.

(i) The motion of an object under the influence of force of gravity alone is called a free fall.
(ii) The acceleration produced in the motion of an object falling freely under the influence of gravity is called acceleration due to gravity.
(iii) The value of 'g' near the surface of the earth is 9.8 m/s².
(iv) The direction of acceleration due to gravity is towards the centre of the earth i.e., in the vertically downward direction.

Buoyant force, u = V p g
As the density p of saturated solution is more than that of water, the cube will experience a greater buoyant force in a saturated solution
\(\frac { Buoyant\quad force\quad on\quad 4\quad cm\quad cube }{ Buoyant\quad force\quad on\quad 5\quad cm\quad cube } =\frac { { V }_{ 2 }\rho \quad g }{ { V }_{ 1 }\rho \quad g } =\frac { { V }_{ 2 } }{ { V }_{ 1 } } ={ \left( \frac { 4 }{ 5 } \right) }^{ 2 }=64:125.\)

If the density of an object is more than the density of the liquid, then it sinks in the liquid. This is because the buoyant force acting on the object is less than the force of gravity. On the other hand, if the density of the object is less than the density of the liquid, then it floats on the surface of the liquid. This is because the buoyant force acting on the object is greater than the force of gravity.

Pressure. The thrust acting per unit area of a called pressure.
Pressure = \(\frac{Force}{Area}\)
If a force F acts normally over an area A of a surface, then the pressure,
The SI unit of pressure is newton per square metre (Nm^-2). It is also called Pascal (Pa).

Importance of the universal law of gravitation. The universal law of gravitation successfully explained many phenomena occurring in nature. Some of these phenomena are as follows:
1. The force that binds us to the earth.
2. The motion of the moon around the earth.
3. The motion of planets around the sun.
4. The tides due to the moon.

Gravitation. Everybody in this universe attracts every other body with a force known as 'force of gravitation'. Gravitation is the force of attraction between any two bodies in the universe. The attraction between the sun the earth, the attraction between a table and a chair lying in a room, the attraction between the earth and a satellite revolving around it etc.; are all examples of gravitation.
Gravity. Gravity is a special case of gravitation. Gravity is the attraction between the earth and any object lying on or near its surface. A body thrown up falls bask on the surface of the earth due to earth's force of gravity.

(i) Since, it is known that a body can sink in water only when its weight is greater than the upthrust act on it by the water. But in this case, the upthrust act on the bottle is greater than its weight, thats why, it bounces back on the water surface.
(ii) A bucket of water weighs less inside the well water, it is because when the bucket immersed in water fully, upthrust act on it by water which reduce its actual weight.

Archimedes' principle. This principle states that when a body is immersed fully or partially in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it.
Applications of Archimedes' principle
(i) Archimedes' principle is used in designing ships and submarines.
(ii) Lactometers based on Archimedes' principle are used to measure purity of a sample of milk.
(iii) Hydrometers used to measure density off liquids are based on Archimedes' principle.

The gravity of the Earth attracts every object towards its centre. When an object is released from a height, it falls towards the surface of the Earth under the influence of gravitational force. The motion of the object is said to have free fall.

Force of gravitation, F = \(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \)
(i) When mass of one body (m1 or m2) is doubled, the force gets doubled.
\(F'=G\frac { { (2m }_{ 1 }){ m }_{ 2 } }{ { r }^{ 2 } } =2G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } =2F\)
(ii) when the distance between the bodies is doubled, 
\(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ (2{ r }^{ 2 }) } =\frac { 1 }{ 4 } G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \frac { 1 }{ 4 } F\)i.e. the force becomes one-fourth of the original force.
(iii) When the masses of both bodies are doubled,
\(F'=G\frac { { (2m }_{ 1 }){ (2m }_{ 2 }) }{ { r }^{ 2 } } =4G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } =4F\)
i.e., the force becomes four times the original force.
(iii) When the distance between the two bodies is tripled,
\(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ (3{ r }^{ 2 }) } =\frac { 1 }{ 9 } G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \frac { 1 }{ 9 } F\)
i.e., the force becomes one-ninth of the originals force.

(i) The relative density of a substance is the ratio of its density to that of water.
Relative density of a substance
= \(\cfrac { Density\quad of\quad the\quad substance }{ Density\quad of\quad water } \)
Relative density of a substance
=\(=\cfrac { Mass\quad of\quad the\quad substance }{ Volume\quad of\quad the\quad substance } \times \cfrac { Volume\quad of\quad water }{ Mass\quad of\quad water } \)  \(\left[ \therefore Density=\cfrac { mass }{ volume } \right] \)
Given that, mass of the cube = 50 g
Side of cube = 1.5 cm
\(\therefore\) Volume of cube = (1.5)³ cm³ = 3.375 cm³
\(\therefore\) Density =\(\cfrac { mass }{ Volume } \)
=\(\cfrac { 50 }{ 3.375 } \) = 14.81 g cm^-3
(iil) Given that, mass, m = 250 g
Volume, V= 400 cc
\(\therefore \) Density=\(\cfrac { mass }{ Volume } \)
= \(\cfrac { 250 }{ 400 } \)g(cc)^-1= 0.625 g (cc)^-1
As we know that, density of water = 1 g (cc)^-1. So, density of tin is less than that of water and hence tin will float.

() Difference between thrust and pressure are

The pressure exerted by 1 N of force, acting perpendicular on the s~rface of 1m 2 area is called 1pascal.
1 Pa= 1Nm^-2
(1)
The force required to accelerate 1 kilogram of mass at the rate of 1 metre per second square is called 1newton.
1N = 1 kg ms^-2
Since, pressure is inversely proportional ro the area 0 contact, Le. \(p\propto \cfrac { 1 }{ A } \)
Therefore, pressure will reduce to half, if area of contact is doubled.

(i) (a) g on the moon is given by
\({ g }^{ ' }=\cfrac { g }{ 6 } =\cfrac { 9.8 }{ 6 } \)
= 1.63 m/s²
(b) Mass of the person on the moon
=\(\cfrac { 110.84 }{ 1.63 } =68kg\)
(c) Weight of person on the earth = mg
= 68 x9.8
= 666.4 N- m²/kg²
(ii) Weight of a person on the earth will be
\(w=\cfrac { GMm }{ { R }^{ 2 } } \)
where, M = mass of the earth
R = radius of the earth
m = mass of person
and G =6.67 X 10 ^-11 N-m²/kg²

(t) Let the two bodies have masses m) and mz and they are placed at the same distance R from the centre of the earth. According to the question, if the sameforce  acts on both of them, then
\({ F }_{ 1 }=\cfrac { GM{ m }_{ 1 } }{ { R }^{ 2 } } \)
and \({ F }_{ 2 }=\cfrac { GM{ m }_{ 2 } }{ { R }^{ 2 } } \)
As, F₁=F₂
Hence,\(\cfrac { GM{ m }_{ 1 } }{ { R }^{ 2 } } =\cfrac { GM{ m }_{ 2 } }{ { R }^{ 2 } } \)
So, m₁ = m₂, their masses will be same
(ii) Mathematically,\(g=\cfrac { GM }{ { R }^{ 2 } } \)
where, g= acceleration due to gravity
G = universal gravitational constant
M = mass of the earth
and R = radius of the earth 
(ii) G is known as the universal gravitational constant because its value remains same all the time everywhere in the universe, applicable to all bodies whether celestial or terrestrial.