Understanding the Problem:

We are given that a street car moves rectilinearly from station A to the next stop B. The acceleration of the street car varies according to the law f = a - bx, where a and b are positive constants, and x is the distance from station A. We need to determine the distance AB.

Solution:

To solve this problem, we will follow these steps:

Step 1: Deriving the Equation of Motion

The equation of motion relates the variables of distance, velocity, acceleration, and time. We need to derive this equation using the given information.

The acceleration of the street car is given as f = a - bx.

Using the definition of acceleration (f = dv/dt), we can write:

a - bx = dv/dt

Rearranging the terms, we get:

dv = (a - bx) dt

Integrating both sides, we get:

∫ dv = ∫ (a - bx) dt

Integrating the left side gives us:

v = at - (bx^2)/2 + C1

Where C1 is the constant of integration.

Step 2: Applying Initial Conditions

We know that when the car is at station A (x = 0), its velocity is zero (v = 0).

Substituting these values into the equation, we get:

0 = a(0) - b(0)^2/2 + C1

0 = C1

So, C1 = 0.

Step 3: Finding the Equation of Distance

Now, we can find the equation of distance (x) by integrating the equation of velocity with respect to time.

dx = v dt

Integrating both sides, we get:

∫ dx = ∫ (at - (bx^2)/2) dt

x = (a/2)t^2 - (b/6)x^3 + C2

Where C2 is the constant of integration.

Step 4: Applying Boundary Condition

When the car reaches the next stop B, the distance x is equal to the distance AB. Let's denote the distance AB as D.

So, when x = D, we have:

D = (a/2)t^2 - (b/6)D^3 + C2

Rearranging the terms, we get:

(b/6)D^3 = (a/2)t^2 + C2 - D

Simplifying further, we get:

(b/6)D^3 = (a/2)t^2 + (6C2 - 6D)

Comparing the above equation, we can see that the left side is a constant (b/6)D^3, and the right side is a function of time (t^2).

Step 5: Determining the Distance AB

Since the left side is a constant and the right side is a function of time, they must be equal for all values of time (t). Therefore, the constant term on the right side must also be equal to the constant term on the left side.

6C2 - 6D = 0

Simplifying, we get:

C2 = D

Substituting this value back into the equation, we get:

(b/