Introduction: In this problem, we have to find the distance travelled by a particle in the first 2 seconds along the x-axis, given its position-time function.

Given:
x = 1 t - t^2

Solution:
To find the distance travelled by the particle, we need to find the total path length covered by the particle in the first 2 seconds. We can do this by finding the magnitude of the displacement vector of the particle over this time interval.

Step 1: Find the velocity of the particle
We know that velocity is the time derivative of the position function. Therefore,
v = dx/dt = 1 - 2t

Step 2: Find the displacement of the particle in the first 2 seconds
To do this, we need to integrate the velocity function over the interval [0,2].
Δx = ∫v dt = ∫0^2 (1 - 2t) dt = t - t^2 [Evaluate the integral using integration by substitution]

Step 3: Find the distance travelled by the particle in the first 2 seconds
The distance travelled by the particle is the magnitude of the displacement vector, which is given by the absolute value of the displacement.
distance = |Δx| = |t - t^2| [Taking absolute value as we are interested in distance]

Step 4: Calculate the distance travelled by the particle in the first 2 seconds
Now, we just need to substitute t = 2 in the above expression to get the distance travelled by the particle in the first 2 seconds.
distance = |2 - 2^2| = 2 units

Conclusion: Therefore, the distance travelled by the particle in the first 2 seconds is 2 units.