Explanation

The degrees of freedom refer to the number of independent ways a gas molecule can store energy.

For a monoatomic gas (e.g., noble gases like helium, neon, argon):

• It has only translational motion along the x, y, and z axes.
• There are no rotational or vibrational degrees of freedom.

Thus, the total degrees of freedom for a monoatomic gas is 3 (one for each spatial direction: x, y, and z).

Explanation

Escape velocity (vₑ) is given by:

vₑ = √(2GM/r)

where:

G = gravitational constant

M = mass of the planet

r = radius of the planet

Given:

M₂ = 1000M₁ (mass of the new planet is 1000 times that of Earth)

r₂ = 10r₁ (radius of the new planet is 10 times that of Earth)

Escape velocity on the new planet (vₑ₂) is:

vₑ₂ = √(2GM₂/r₂)

= √(2G(1000M₁)/(10r₁))

= √(200G(M₁/r₁))

= √200 × √(2GM₁/r₁)

= √200 × vₑ₁

Given vₑ₁ = 11.2 km/s:

vₑ₂ ≈ √200 × 11.2 km/s

≈ 112 km/s

Explanation

Explanation

According to Kepler's third law:

The square of the time period (T²) of a planet.

The Sun is directly proportional to the cube of the semi-major axis (R³) of its orbit.

Mathematically, this can be expressed as:

T² ∝ R³

Explanation

When a proton and anti-proton annihilate, the energy release is equal to the sum of their rest mass energies.

The rest mass energy of a proton (or anti-proton) is approximately:

E = mc²

= (1.67 × 10⁻²⁷ kg) × (3 × 10⁸ m/s)²

≈ 1.5 × 10⁻¹⁰ J

Since both the proton and anti-proton annihilate, the total energy release is:

E_total = 2 × 1.5 × 10⁻¹⁰ J

≈ 3 × 10⁻¹⁰ J

Explanation

Momentum (p) is proportional to velocity (v), while kinetic energy (KE) is proportional to v².

When momentum decreases by 20%, velocity decreases by 20%.

So, kinetic energy decreases by (1 - 0.8²) × 100% ≈ 36%.

Explanation

Let's denote the weight of the man in the stationary lift as W and the weight in the accelerating lift as W'.

We know that W' / W = 2/3.

When the lift is accelerating downward with acceleration 'a', the apparent weight (W') is given by:

W' = W - ma

where m is the mass of the man.

Since W' / W = 2/3, we can write:

(W - ma) / W = 2/3

Substitute W = mg (where g is the acceleration due to gravity):

(mg - ma) / mg = 2/3

Simplify:

g - a = (2/3)g

a = g - (2/3)g

a = (1/3)g

a = g/3

Explanation

For maximum range, the angle of projection should be 45°.

However, since that's not an option, we need to find the angle closest to 45°.
Among the given options:

• - 35° is closer to 45° than 20°.
• - 60° is closer to 45° than 70°.

However, 35° is closer to 45° than 60°.
So, the correct answer is: 35°.

Explanation

Since the smaller part becomes stationary, it means that the larger part will carry all the momentum.

Let's say the mass of the smaller part is m and the mass of the larger part is 3m.

Initial momentum = Final momentum

mV + 3mV = 3mV'

where V' is the velocity of the larger part.

Combine like terms:

4mV = 3mV'

Divide both sides by 3m:

V' = 4V/3

Explanation

When a man throws a ball to the maximum height (h), the initial velocity (v₀) is entirely vertical.

The maximum height (h) is given by:

h = v₀² / (2g)

where g is the acceleration due to gravity.

Now, when the man throws the same ball to the maximum horizontal distance, the initial velocity (v₀) is entirely horizontal.

The maximum horizontal distance (R) is given by:

R = v₀² / g

Substituting the expression for v₀² from the first equation:

R = 2h