Analysis:

To calculate the time taken to cover the first half and the second half of the run, we need to break down the sprinter's motion into two parts: the first three-quarters of the total run, where she increases her speed uniformly, and the last quarter, where she covers the distance with a constant speed.

First Three-Quarters of the Run:

Let's assume the total distance of the sprint is 100m. The first three-quarters of the run covers 75m. We are given that the sprinter increases her speed uniformly at a rate of 1.5m/s^2.

Using the Kinematic Equation:
v^2 = u^2 + 2as

Where:
v = final velocity
u = initial velocity
a = acceleration
s = distance

We can rearrange the equation to solve for time:

t = (v - u) / a

Initial Velocity:
The initial velocity is zero since the sprinter starts from rest.

u = 0 m/s

Final Velocity:
To find the final velocity, we need to calculate the acceleration during the first three-quarters of the run.

a = 1.5 m/s^2

We know that the sprinter covers a distance of 75m during this time, and the acceleration is uniform. Using the equation:

s = ut + (1/2)at^2

We can rearrange the equation to solve for the final velocity:

v = u + at

v = 0 + 1.5 * t

v = 1.5t

Since the final velocity at the end of the three-quarters run is 1.5t m/s, we can substitute this value into the time equation:

t = (1.5t - 0) / 1.5

t = t

This means the time taken to cover the first three-quarters of the run is equal to the final velocity divided by the acceleration:

t = 1.5 / 1.5

t = 1 second

Therefore, the sprinter takes 1 second to cover the first three-quarters of the run.

Last Quarter of the Run:

The last quarter of the run covers a distance of 25m. We are given that the sprinter covers this distance with a uniform speed.

Since the speed is constant and no acceleration is involved, we can use the formula:

t = s / v

Where:
t = time
s = distance
v = speed

Given that the distance is 25m, and the speed is uniform, we can calculate the time taken to cover the last quarter of the run:

t = 25 / v

However, we do not have the value of the speed (v) during this part of the run. Without this information, we cannot determine the exact time taken to cover the last quarter of the run.

Conclusion:

In conclusion, the sprinter takes 1 second to cover the first three-quarters of the run. However, without knowing the speed during the last quarter, we cannot determine the exact time taken to cover it.