The given function for acceleration, a(t) = bt, indicates that the acceleration of the particle is directly proportional to time. The constant b determines the rate at which the acceleration increases with time.



The particle starts from the origin, which means its initial position is x = 0. It also has an initial velocity of v.



To determine the distance traveled by the particle in time t, we need to find its position as a function of time. This can be done by integrating the acceleration function with respect to time twice.

First, integrating the acceleration function will give us the particle's velocity as a function of time:

v(t) = ∫a(t) dt = ∫bt dt = (1/2)bt^2 + C1,

where C1 is the constant of integration.

Integrating the velocity function will give us the particle's position as a function of time:

x(t) = ∫v(t) dt = ∫((1/2)bt^2 + C1) dt = (1/6)bt^3 + C1t + C2,

where C2 is another constant of integration.



To determine the values of the constants C1 and C2, we can use the initial conditions given in the problem.

Since the particle starts from the origin, x(0) = 0. Plugging this into the position function:

x(0) = (1/6)b(0)^3 + C1(0) + C2 = 0.

This implies that C2 = 0.

The initial velocity is given as v. Plugging this into the velocity function:

v(0) = (1/2)b(0)^2 + C1 = v.

This implies that C1 = v.



Using the determined values of C1 and C2, the position function becomes:

x(t) = (1/6)bt^3 + vt.



To find the distance traveled by the particle in time t, we need to consider the absolute value of the position function, as distance is always positive:

|x(t)| = |(1/6)bt^3 + vt|.

As this equation represents the magnitude of the position, it gives the distance traveled by the particle in time t. The distance can be positive or negative depending on the direction of motion.



In summary, the distance traveled by the particle starting from the origin with an initial velocity v, and with acceleration increasing as (bt), is given by the equation |(1/6)bt^3 + vt|. The acceleration function is integrated to find the velocity function, and the velocity function is integrated to find the position function. The constants of integration are determined by applying the initial conditions. The resulting position function represents the particle's position as a function of time, and the absolute value of this function gives the distance traveled.