A particle located at x = 0 at time t = 0, starts moving along the positive x–direction with a velocity 'v' which varies as , then velocity of particle varies with time as (α is a constant) :

The given condition states that the velocity 'v' of the particle varies with the position 'x'. This can be written as:

v = αx

However, we know from physics that velocity is also the rate of change of displacement with respect to time. This can be written as:

v = dx/dt

Equating the two equations, we get:

dx/dt = αx

This is a first order linear differential equation. The solution to this equation gives the relationship between the position 'x' and time 't'.

To solve this, we can separate the variables and integrate:

∫dx/x = ∫α dt

On integrating, we get:

ln|x| = αt + C

where C is the integration constant.

Exponentiating both sides to get rid of the natural logarithm, we get:

|x| = e^(αt+C)

Since the particle starts at x = 0 at t = 0, we can find the value of C by substituting these values into the equation. This gives C = 0.

So, the final equation describing how the velocity of the particle varies with time is:

v(t) = αe^(αt)