To understand why the correct answer is option 'C', let's break down the problem into different parts:

1. Orbital Speed:
The orbital speed of a satellite near the Earth's surface is the speed at which it needs to move horizontally to maintain a stable orbit around the Earth. It depends on the gravitational force between the satellite and the Earth. The formula for orbital speed is given by:

v = √(GM/R)

Where:
- v is the orbital speed
- G is the gravitational constant
- M is the mass of the Earth
- R is the radius of the Earth

2. Projecting a Particle Upward:
When a particle is projected upward from the Earth's surface with a speed equal to the orbital speed, it means that the initial velocity of the particle is equal to the orbital speed (v).

3. Height to which it would rise:
To find the height to which the particle would rise, we need to consider the motion of the particle under the influence of gravity. The particle will experience a deceleration due to gravity until it reaches its maximum height and then start to fall back down to the Earth.

At its maximum height, the particle's velocity will be zero (v = 0) because it comes to a stop before reversing its direction. Using the equation of motion:

v^2 = u^2 + 2as

Where:
- v is the final velocity (zero at maximum height)
- u is the initial velocity (equal to the orbital speed)
- a is the acceleration due to gravity (constant)
- s is the displacement (height)

We can rearrange the equation to solve for the displacement:

s = (v^2 - u^2) / (2a)

Since v = 0, the equation simplifies to:

s = -u^2 / (2a)

4. Applying the equations:
Now we can substitute the values into the equation to find the height (s). The negative sign indicates that the displacement is in the opposite direction to the initial velocity.

s = -(v^2) / (2a)

Substituting the value of u = v (since the initial velocity is equal to the orbital speed):

s = -(v^2) / (2a)

s = -(v^2) / (2 * (-g))

s = v^2 / (2g)

Where g is the acceleration due to gravity.

5. Simplifying the equation:
To simplify the equation further, let's substitute the value of g using the formula:

g = GM / R^2

s = v^2 / (2 * (GM / R^2))

s = v^2 * (R^2 / (2GM))

s = v^2 * (R / (2G))

6. Conclusion:
From the equation s = v^2 * (R / (2G)), we can see that the height (s) to which the particle would rise is directly proportional to the radius of the Earth (R). Therefore, the correct answer is option 'C' - the height to which the particle would rise is R.