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 will follow the logic presented in the video transcript while breaking it down into clear steps.

### Step 1: Understand the given information
We are given:
- Initial speed of the particle, \( u \)
- Retardation (deceleration), \( a \)
- Time taken to come to rest, \( T \)

### Step 2: Find the total distance traveled
Using the equation of motion, we can find the total distance \( s \) traveled by the particle before it comes to rest:
\[ s = uT - \frac{1}{2} a T^2 \]
Here, \( s \) is the total distance traveled.

### Step 3: Express the distance for half the path
We need to find the time taken to cover the first half of the total distance, which is \( \frac{s}{2} \). We can express this as:
\[ \frac{s}{2} = u t - \frac{1}{2} a t^2 \]
where \( t \) is the time taken to cover half the distance.

### Step 4: Substitute \( s \) in the half distance equation
Substituting \( s \) from Step 2 into the equation for \( \frac{s}{2} \):
\[ \frac{1}{2} \left( uT - \frac{1}{2} a T^2 \right) = u t - \frac{1}{2} a t^2 \]
This simplifies to:
\[ \frac{uT}{2} - \frac{1}{4} a T^2 = u t - \frac{1}{2} a t^2 \]

### Step 5: Rearranging the equation
Rearranging this equation gives us:
\[ \frac{1}{2} a t^2 - u t + \left( \frac{uT}{2} - \frac{1}{4} a T^2 \right) = 0 \]
This is a quadratic equation in terms of \( t \).

### Step 6: Apply the quadratic formula
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we identify:
- \( a = \frac{1}{2} a \)
- \( b = -u \)
- \( c = \frac{uT}{2} - \frac{1}{4} a T^2 \)

Substituting these into the quadratic formula:
\[ t = \frac{-(-u) \pm \sqrt{(-u)^2 - 4 \cdot \frac{1}{2} a \left( \frac{uT}{2} - \frac{1}{4} a T^2 \right)}}{2 \cdot \frac{1}{2} a} \]
This simplifies to:
\[ t = \frac{u \pm \sqrt{u^2 - 2a \left( \frac{uT}{2} - \frac{1}{4} a T^2 \right)}}{a} \]

### Step 7: Simplifying further
After simplifying the expression under the square root and solving, we find two potential solutions for \( t \). However, since time cannot be negative, we only consider the positive root.

### Step 8: Final expression for \( t \)
After further simplifications, we arrive at:
\[ t = T \left( 1 - \frac{1}{\sqrt{2}} \right) \]
This gives us the time taken to cover the first half of the total distance traveled.

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