Introduction:
When a particle is projected into the air, it follows a curved trajectory known as a projectile motion. The range of a projectile refers to the horizontal distance it covers before hitting the ground, while the vertical height refers to the maximum height attained during its flight. In this scenario, the range and vertical height are equal, which leads to a specific angle of projection.

Explanation:
To determine the angle of projection, let's consider the following factors:

1. Range:
The range of a projectile can be calculated using the formula:

Range = (v^2 * sin(2θ)) / g

Where:
- v is the initial velocity of the projectile
- θ is the angle of projection
- g is the acceleration due to gravity (approximately 9.8 m/s^2)

2. Vertical height:
The maximum height attained by the projectile can be calculated using the formula:

Vertical height = (v^2 * sin^2(θ)) / (2g)

Equalizing range and vertical height:
Since the range and vertical height are equal, we can equate the two formulas:

(v^2 * sin(2θ)) / g = (v^2 * sin^2(θ)) / (2g)

Simplifying the equation, we get:

2 * sin(θ) * cos(θ) = sin^2(θ)

Using the trigonometric identity sin(2θ) = 2 * sin(θ) * cos(θ), we can rewrite the equation as:

sin(2θ) = sin^2(θ)

Applying the double-angle formula for sine, sin(2θ) = 2 * sin(θ) * cos(θ), we have:

2 * sin(θ) * cos(θ) = sin^2(θ)

Dividing both sides by sin(θ), we get:

2 * cos(θ) = sin(θ)

Dividing both sides by cos(θ), we get:

2 = tan(θ)

Taking the inverse tangent of both sides, we find:

θ = tan^(-1)(2)

Conclusion:
Therefore, the angle of projection for the particle to achieve equal range and vertical height is approximately 63.4 degrees.