The given question is related to the transfer function of a unity feedback system. Let's break down the question and solve it step by step.

1. Given information:
- The system has a unity feedback configuration, which means the output of the system is fed back to the input with a gain of 1.
- The system has a 10% overshoot, which indicates the response of the system when it is subjected to a step input.
- The velocity error constant, Kv, is given as 100.

2. Transfer function of a unity feedback system:
In a unity feedback system, the transfer function is given by:

G(s) = Gp(s) / (1 + Gp(s)H(s))

Where Gp(s) is the transfer function of the plant and H(s) is the transfer function of the controller.

3. Overshoot and damping ratio:
The overshoot is a measure of the response of the system to a step input. It is given by the formula:

Overshoot = e^(-ζπ / √(1 - ζ^2)) * 100

Where ζ is the damping ratio. In this case, the overshoot is given as 10%. By rearranging the formula and substituting the given value, we can calculate the damping ratio:

10 = e^(-ζπ / √(1 - ζ^2))
ln(0.1) = -ζπ / √(1 - ζ^2)
(0.1)^2 = ζ^2π^2 / (1 - ζ^2)
0.01 - ζ^2 = ζ^2π^2
π^2ζ^2 + ζ^2 = 0.01
ζ^2(π^2 + 1) = 0.01
ζ^2 = 0.01 / (π^2 + 1)
ζ = √(0.01 / (π^2 + 1))

4. Calculation of the value of a:
The value of a is related to the damping ratio and overshoot. It is given by the formula:

a = ζ * ωn / √(1 - ζ^2)

Where ωn is the natural frequency of the system. Since the system has a 10% overshoot, we can calculate the value of a using the damping ratio we found earlier.

Substituting the values:

a = √(0.01 / (π^2 + 1)) * ωn / √(1 - (0.01 / (π^2 + 1)))
= √(0.01 / (π^2 + 1)) * ωn / √((π^2 + 1) / (π^2 + 1) - 0.01)
= √(0.01 / (π^2 + 1)) * ωn / √((π^2 + 1 - 0.01) / (π^2 + 1))
= √(0.01 / (π^2 + 1)) * ωn / √(π^2 / (π^2 + 1))
= √(0.01 / (π^2 + 1)) *