Velocity of a body moving a straight line with uniform acceleration (a) reduces by `(3)/(4)` of its initial velocity in time `t_(0)`. The total time of motion of the body till its velocity becomes zero is

Options

A. `(4)/(3)t_(0)`
B. `(3)/(2)t_(0)`
C. `()/(3)t_(0)`
D. `(8)/(3)t_(0)`

Explanation

To solve the problem, we need to determine the total time of motion of the body until its velocity becomes zero, given that its velocity reduces by \( \frac{3}{4} \) of its initial velocity in time \( t_0 \).

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Let the initial velocity of the body be \( u \).
- The final velocity after time \( t_0 \) is given to be \( v = u - \frac{3}{4}u = \frac{1}{4}u \).

2. **Using the Equation of Motion**:
- We can use the first equation of motion:
\[ v = u + at \]
- Substituting the known values:
\[ \frac{1}{4}u = u + a t_0 \]

3. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ a t_0 = \frac{1}{4}u - u = -\frac{3}{4}u \]
- Therefore, we can express acceleration \( a \) as:
\[ a = -\frac{3}{4} \frac{u}{t_0} \]

4. **Finding Total Time Until Velocity Becomes Zero**:
- Now, we need to find the total time \( T \) until the velocity becomes zero. We can use the same equation of motion:
\[ 0 = u + aT \]
- Substituting for \( a \):
\[ 0 = u - \frac{3}{4} \frac{u}{t_0} T \]

5. **Rearranging to Solve for \( T \)**:
- Rearranging gives:
\[ \frac{3}{4} \frac{u}{t_0} T = u \]
- Dividing both sides by \( u \) (assuming \( u \neq 0 \)):
\[ \frac{3}{4} \frac{T}{t_0} = 1 \]
- Therefore:
\[ T = \frac{4}{3} t_0 \]

6. **Final Answer**:
- The total time of motion of the body until its velocity becomes zero is:
\[ T = \frac{4}{3} t_0 \]