A particle starts with initial speed u and retardation a to come to rest in time T. The time taken to cover first half of the total path travelled is

Options

A. `(T)/(sqrt2)`
B. `T(1-(1)/(sqrt2))`
C. `(T)/(2)`
D. `(3T)/(4)`

Answer

To solve the problem step by step, we need to find the time taken to cover the first half of the total distance traveled by a particle that starts with an initial speed \( u \) and comes to rest with a retardation \( a \) in time \( T \).

### Step 1: Determine the total distance traveled before coming to rest.
Using the equation of motion:
\[ v^2 = u^2 + 2as \]
where \( v = 0 \) (final velocity when the particle comes to rest), \( u \) is the initial speed, and \( a = -a \) (retardation). Rearranging gives:
\[ 0 = u^2 - 2as \implies s = \frac{u^2}{2a} \]
Thus, the total distance \( s \) traveled before coming to rest is:
\[ s = \frac{u^2}{2a} \]

### Step 2: Find the distance for the first half of the total path.
The distance for the first half of the total path is:
\[ \frac{s}{2} = \frac{u^2}{4a} \]

### Step 3: Use the equation of motion to find the time taken to cover this distance.
We use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting \( s = \frac{u^2}{4a} \):
\[ \frac{u^2}{4a} = ut - \frac{1}{2} a t^2 \]
Rearranging gives:
\[ \frac{1}{2} a t^2 - ut + \frac{u^2}{4a} = 0 \]

### Step 4: Solve the quadratic equation for \( t \).
This is a quadratic equation in the form \( Ax^2 + Bx + C = 0 \), where:
- \( A = \frac{1}{2} a \)
- \( B = -u \)
- \( C = \frac{u^2}{4a} \)

Using the quadratic formula \( t = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
\[ t = \frac{u \pm \sqrt{u^2 - 2a \cdot \frac{u^2}{4a}}}{a} \]
Simplifying the term under the square root:
\[ t = \frac{u \pm \sqrt{u^2 - \frac{u^2}{2}}}{a} = \frac{u \pm \sqrt{\frac{u^2}{2}}}{a} = \frac{u \pm \frac{u}{\sqrt{2}}}{a} \]
This gives two possible values for \( t \):
\[ t_1 = \frac{u(1 - \frac{1}{\sqrt{2}})}{a} \quad \text{and} \quad t_2 = \frac{u(1 + \frac{1}{\sqrt{2}})}{a} \]

### Step 5: Choose the correct value of time.
Since we are interested in the time taken to cover the first half of the distance, we select the smaller value:
\[ t = \frac{u(1 - \frac{1}{\sqrt{2}})}{a} \]
We can express \( \frac{u}{a} \) as \( T \) (the total time taken to stop):
\[ t = T \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Final Answer:
The time taken to cover the first half of the total path is:
\[ t = T \left(1 - \frac{1}{\sqrt{2}}\right) \]