Introduction:
In this problem, we need to find the distance travelled by a particle in the fifth half second with an acceleration of 2m/s^2.

Formula:
The distance travelled by a particle with constant acceleration can be calculated using the following formula:

S = ut + 1/2 at^2

where
S = distance travelled
u = initial velocity (assumed to be zero)
a = acceleration
t = time

Explanation:
In the given problem, the acceleration of the particle is 2m/s^2. Since the initial velocity is zero, u=0. We need to find the distance travelled by the particle in the fifth half second, which means t=0.5s.

Now, substituting the values in the formula, we get:

S = 0 + 1/2 * 2 * (0.5)^2
S = 0 + 0.5
S = 0.5m

Therefore, the distance travelled by the particle in the fifth half second is 0.5m.

Conclusion:
In conclusion, we can find the distance travelled by a particle with constant acceleration using the formula S = ut + 1/2 at^2. In this problem, we found the distance travelled by a particle with an acceleration of 2m/s^2 in the fifth half second to be 0.5m.

Distance travelled in 5th half second = Distance travelled in 2.5 seconds - Distance travelled in 2 seconds  = 0.5a (t₁² - t₂²) = 0.5 * 2 * (2.5² - 2²) = 2.25m