Derivation:

Let's consider the motion of the train in two phases:
1. Acceleration phase:
During this phase, the train starts from rest and accelerates at a constant rate alpha for a distance x1 in time t1.

2. Deceleration phase:
After reaching the distance x1, the train decelerates at a constant rate beta to come to rest again in time t2, covering a distance x2.

Using the equations of motion:

We can use the equations of motion to derive the relationship between the given variables.

1. Acceleration phase:
Since the train starts from rest, its initial velocity u1 is 0 m/s. The final velocity v1 can be calculated using the equation of motion:
v1 = u1 + alpha * t1

The distance covered during acceleration phase x1 can be calculated using the equation of motion:
x1 = u1 * t1 + (1/2) * alpha * t1^2

2. Deceleration phase:
During the deceleration phase, the train comes to rest, so its final velocity v2 is 0 m/s. The initial velocity u2 can be calculated using the equation of motion:
v2 = u2 + beta * t2

The distance covered during the deceleration phase x2 can be calculated using the equation of motion:
x2 = u2 * t2 + (1/2) * beta * t2^2

Deriving the relationship:

To prove x1/x2 = t1/t2 = beta/alpha, we need to show that these ratios are equal.

1. x1/x2:
Dividing the equation for x1 by the equation for x2, we get:
x1/x2 = (u1 * t1 + (1/2) * alpha * t1^2) / (u2 * t2 + (1/2) * beta * t2^2)

Since u1 = 0 and u2 = v1, we can simplify further:
x1/x2 = (1/2) * alpha * t1^2 / (v1 * t2 + (1/2) * beta * t2^2)

Using the equation v1 = alpha * t1, we get:
x1/x2 = (1/2) * alpha * t1^2 / (alpha * t1 * t2 + (1/2) * beta * t2^2)

Cancelling out alpha and rearranging, we get:
x1/x2 = (1/2) * t1 / (t1 * t2 / t2 + (1/2) * beta * t2 / alpha)
x1/x2 = (1/2) * t1 / (t1 + (1/2) * beta * t2 / alpha)

Since v1 = alpha * t1, we can substitute it back into the equation:
x1/x2 = (1/2) * t1 / (t1 + (1/2) * beta * t2 / v1)

2. t1/t2:
Dividing the equation for t1 by the equation for t2, we get:
t1/t2 = (v1 + alpha * t1) / (u2 + beta * t2)

Since u