**Velocity as a Function of Time**

To determine the velocity attained by a particle starting from rest and experiencing variable acceleration described by the equation f = kt, we can integrate the equation to obtain the velocity as a function of time.

**Integration of Acceleration Function**

The given equation f = kt represents the variable acceleration experienced by the particle. Here, f represents the force acting on the particle, k is a constant, and t is the time.

We know that acceleration is the rate of change of velocity with respect to time. Therefore, we can write:

a = dv/dt,

where a is the acceleration and v is the velocity.

Since the force acting on the particle is given by f = kt, we can substitute this into the equation for acceleration:

kt = dv/dt.

**Integration Process**

To find the velocity as a function of time, we can integrate both sides of the equation with respect to time:

∫kt dt = ∫dv.

Integrating both sides gives:

(1/2)kt^2 + C1 = v,

where C1 is the constant of integration.

**Applying Initial Conditions**

Given that the particle starts from rest, its initial velocity v(0) = 0. We can substitute this condition into the equation:

(1/2)k(0)^2 + C1 = 0,
C1 = 0.

Thus, the equation becomes:

(1/2)kt^2 = v.

**Velocity as a Proportional Function**

From the above equation, it is clear that the velocity attained at the end of time t is directly proportional to the square of time. As time increases, the velocity of the particle increases as well.

Therefore, the velocity attained at the end of time t is proportional to t^2.

T2