Explanation:

When light of frequency twice the threshold frequency is incident on the metal plate, the maximum velocity of emitted electron is v1. When the frequency of incident radiation is increased to five times the threshold value, the maximum velocity of emitted electron becomes v2. We need to find the value of x, where v2 = x * v1.

Threshold Frequency:
The threshold frequency is the minimum frequency of light required to emit electrons from a metal surface. Below the threshold frequency, no electrons are emitted, regardless of the intensity of the incident light.

Effect of Incident Light Frequency on Electron Kinetic Energy:
The kinetic energy of an electron emitted from a metal surface due to the photoelectric effect depends on the frequency of the incident light. When the frequency of the incident light is increased, the maximum kinetic energy of the emitted electrons also increases.

Relation Between Maximum Kinetic Energy and Maximum Velocity:
The maximum kinetic energy (Kmax) of an emitted electron is given by the equation:
Kmax = (1/2)mv^2
where m is the mass of the electron and v is its velocity.

Relation Between Incident Light Frequency and Maximum Kinetic Energy:
The maximum kinetic energy of an emitted electron is directly proportional to the frequency of the incident light. Mathematically, this can be represented as:
Kmax ∝ f
where f is the frequency of the incident light.

Relation Between Maximum Velocity and Maximum Kinetic Energy:
The maximum velocity (v) of an emitted electron can be derived from its maximum kinetic energy (Kmax) using the equation:
Kmax = (1/2)mv^2
Solving for v:
v = sqrt((2Kmax)/m)
This equation shows that the maximum velocity of an emitted electron is directly proportional to the square root of its maximum kinetic energy.

Relation Between Incident Light Frequency and Maximum Velocity:
Combining the above relations, we can conclude that the maximum velocity of an emitted electron is directly proportional to the square root of the frequency of the incident light. Mathematically, this can be represented as:
v ∝ sqrt(f)

Given Information:
1. When light of frequency twice the threshold frequency is incident on the metal plate, the maximum velocity of emitted electron is v1.
2. When the frequency of incident radiation is increased to five times the threshold value, the maximum velocity of emitted electron becomes v2.

Relationship Between v1 and v2:
From the above information, we can determine the relationship between v1 and v2. Since the frequency of the incident light is directly proportional to its velocity, we can write:
v1 ∝ sqrt(2f0) (where f0 is the threshold frequency)
v2 ∝ sqrt(5f0)

To find the relationship between v2 and v1, we can divide the equations:
v2/v1 = (sqrt(5f0))/(sqrt(2f0))
v2/v1 = sqrt(5f0/2f0)
v2/v1 = sqrt(5/2)
v2/v1 = sqrt(2.5)

Thus, the value of x, where v2 = x * v1, is 2.5.