To calculate the spaceship's acceleration, we first need to know its speed relative to an observer. Let's assume that the spaceship is traveling at 0.9 times the speed of light (0.9c).

The relativistic acceleration formula is given by:

a = (γ^3 * a0)/(1 + γ^3 * a0 * t/c)

where:
a is the spaceship's acceleration
γ is the Lorentz factor given by γ = 1/√(1 - (v^2/c^2))
a0 is the proper acceleration (acceleration from the spaceship's frame of reference)
t is the proper time (time measured by the spaceship)
c is the speed of light

Since the spaceship is traveling at relativistic speeds and time dilation occurs, the proper acceleration (a0) and proper time (t) can't be directly measured by an observer outside the spaceship.

Let's assume that the proper acceleration of the spaceship is a0 = 1 g = 9.8 m/s^2 (acceleration due to gravity on Earth).

To calculate the acceleration, we need to calculate the Lorentz factor γ:
γ = 1/√(1 - (v^2/c^2)) = 1/√(1 - (0.9^2/1^2)) ≈ 2.294

Now we can calculate the spaceship's acceleration:
a = (2.294^3 * 9.8)/(1 + 2.294^3 * 9.8 * t/c)

Since we don't have a specific time measurement, we can't calculate the exact acceleration. However, we can observe that as t approaches infinity, the acceleration approaches a0 (the proper acceleration). Hence, we can assume the spaceship's acceleration is approximately 9.8 m/s^2.

To calculate the distance the spaceship can safely travel before a collision becomes likely, we need to consider the time it takes for a collision to become likely and the spaceship's speed. Let's assume that the time until a collision becomes likely is 1 hour.

Distance = (1/2) * a * t^2 = (1/2) * 9.8 * (3600^2)

Distance ≈ 63,072,000 meters or approximately 63,072 kilometers

Therefore, the spaceship can safely travel about 63,072 kilometers through the dense asteroid field before a collision becomes likely.

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