Given:
- De Broglie wavelength of particle B is double that of particle A.
- Mass of particle A is three times that of particle B.

To find:
The ratio of velocity of particle A to particle B.

Solution:
The de Broglie wavelength is given by the equation:

λ = h / p

where λ is the de Broglie wavelength, h is the Planck's constant, and p is the momentum of the particle.

Step 1: Comparing de Broglie wavelengths

Let the de Broglie wavelength of particle A be λA and that of particle B be λB.

According to the given information, λB is double that of λA.

λB = 2 * λA

Step 2: Comparing masses

Let the mass of particle A be mA and that of particle B be mB.

According to the given information, mA is three times that of mB.

mA = 3 * mB

Step 3: Comparing velocities

The momentum of a particle can be calculated using the equation:

p = mv

where p is the momentum, m is the mass, and v is the velocity of the particle.

The de Broglie wavelength can also be expressed in terms of momentum:

λ = h / p

Comparing the de Broglie wavelengths of particles A and B:

λA = h / pA
λB = h / pB

Substituting the values of momentum:

λA = h / (mA * vA)
λB = h / (mB * vB)

Step 4: Finding the ratio of velocities

Substituting the values of λB and λA from Step 1:

2 * λA = λB
2 * (h / (mA * vA)) = h / (mB * vB)

Simplifying the equation:

2 * mA * vA = mB * vB

Dividing both sides by mB * vA:

(2 * mA * vA) / (mB * vA) = vB / vA

Simplifying further:

2 * mA / mB = vB / vA

Since mA = 3 * mB (from Step 2):

2 * (3 * mB) / mB = vB / vA

6 = vB / vA

Therefore, the ratio of the velocities of particle A to particle B is 6:1.