Time taken to cover the first half of the total path length travelled

To determine the time taken to cover the first half of the total path length travelled by a particle with initial speed u and retardation a, we need to understand the motion of the particle and use the relevant equations of motion.

Understanding the motion
1. The particle is initially moving with a speed u.
2. It experiences a retardation a, which causes it to slow down.
3. Eventually, the particle comes to rest in time T.

Equations of motion
1. The equation of motion for the particle's velocity as a function of time is: v = u - at, where v is the final velocity, u is the initial velocity, a is the retardation, and t is the time taken.
2. The equation of motion for the particle's displacement as a function of time is: s = ut - 0.5at^2, where s is the displacement.

Calculating the time taken to come to rest
Using the equation of motion for velocity, we can set v = 0 (since the particle comes to rest) and solve for t:
0 = u - at
at = u
t = u/a

Therefore, the time taken for the particle to come to rest is t = u/a.

Calculating the time taken to cover the first half of the total path length
1. The total path length travelled by the particle can be calculated using the equation of motion for displacement:
s = ut - 0.5at^2
2. Since we are interested in the time taken to cover the first half of the total path length, we can consider the displacement at that point to be half of the total displacement.
s/2 = ut/2 - 0.5at^2/2
s/2 = ut/2 - 0.5at^2/2
s/2 = ut/2 - 0.25at^2
s/2 = (u/2)t - 0.25at^2

Now, we need to find the time taken to cover this distance. Let's call it t1.

3. Substituting the value of t1 in the above equation, we get:
s/2 = (u/2)t1 - 0.25at1^2

4. Rearranging the equation, we get a quadratic equation:
0.25at1^2 - (u/2)t1 + (s/2) = 0

5. Solving this quadratic equation using the quadratic formula, we can find the value of t1.

Summary
The time taken to cover the first half of the total path length travelled by the particle can be calculated by solving the quadratic equation derived from the equation of motion for displacement.