Motion of a Particle along the x-axis:
The motion of a particle along the x-axis is given by the equation x = 28t - 4t^2. We are required to find the distance traveled by the particle from t = 0 to t = 2.

Calculating the Distance Traveled:
To find the distance traveled by the particle, we need to calculate the displacement (change in position) and take its absolute value.

Step 1: Calculate the Displacement:
The displacement of the particle can be found by subtracting its initial position (at t = 0) from its final position (at t = 2).
- Initial position: x(0) = 28(0) - 4(0^2) = 0
- Final position: x(2) = 28(2) - 4(2^2) = 56 - 16 = 40

The displacement is given by the difference in positions: Δx = x(2) - x(0) = 40 - 0 = 40.

Step 2: Calculate the Distance:
The distance traveled by the particle is the absolute value of the displacement. Since distance cannot be negative, we take the absolute value of the displacement.

Distance = |Δx| = |40| = 40 units.

The particle has traveled a distance of 40 units from t = 0 to t = 2 along the x-axis.

Summary:
- Motion of the particle along the x-axis is described by the equation x = 28t - 4t^2.
- To find the distance traveled, we calculate the displacement by finding the difference in positions at t = 0 and t = 2.
- The displacement is Δx = 40 units.
- Taking the absolute value of the displacement, we find that the particle has traveled a distance of 40 units along the x-axis from t = 0 to t = 2.

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