Introduction:
Weight is the force exerted by an object due to gravity. It is directly proportional to the mass of the object. In this question, we are given that a man weighs 600N at the surface of the Earth and we need to find the height above the surface of the Earth at which he will weigh 300N.

Understanding the Problem:
To solve this problem, we need to know the relationship between weight and height above the Earth's surface. As we move away from the surface of the Earth, the gravitational force acting on an object decreases. Therefore, the weight of an object decreases as we move higher above the surface.

Solution:
We can solve this problem by using the concept of gravitational force and the universal law of gravitation.

Step 1: Finding the gravitational force on the surface of the Earth:
The weight of the man on the surface of the Earth is given as 600N. This is the force acting on the man due to gravity. We know that weight is given by the equation:

Weight = mass × acceleration due to gravity

Since the acceleration due to gravity on the surface of the Earth is constant and equal to 9.8 m/s², we can rearrange the equation to find the mass of the man:

Mass = Weight ÷ acceleration due to gravity
Mass = 600N ÷ 9.8 m/s²
Mass ≈ 61.22 kg

Step 2: Using the universal law of gravitation:
The universal law of gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Mathematically, it can be represented as:

F = (G × m1 × m2) / r²

Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67 × 10^-11 Nm²/kg²)
m1 and m2 are the masses of the two objects
r is the distance between their centers

In this case, the two objects are the man and the Earth. We know the mass of the man (61.22 kg) and we need to find the distance (height) at which the gravitational force becomes 300N.

Step 3: Finding the height at which the man weighs 300N:
We can rearrange the equation for gravitational force to solve for the distance (height):

r² = (G × m1 × m2) / F
r² = (6.67 × 10^-11 Nm²/kg² × 61.22 kg × 5.98 × 10^24 kg) / 300N
r² ≈ 2.678 × 10^7 m²
r ≈ √(2.678 × 10^7) m
r ≈ 5177.3 m

Therefore, the man will weigh 300N at a height of approximately 5177.3 meters above the surface of the Earth.

Conclusion:
In conclusion, a man will weigh 300N at a height of approximately 5177.3 meters above the surface of the Earth. This can be calculated by using the concepts of weight, gravitational force, and the universal law of gravitation.