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 will follow these steps:

### Step 1: Understand the given information
We have a body moving in a straight line with an initial velocity \( v_0 \) and uniform acceleration \( a \). The velocity reduces by \( \frac{3}{4} \) of its initial velocity in time \( t_0 \).

### Step 2: Determine the final velocity after time \( t_0 \)
The reduction in velocity is \( \frac{3}{4} v_0 \). Therefore, the final velocity \( v \) after time \( t_0 \) can be calculated as:
\[ v = v_0 - \frac{3}{4} v_0 = \frac{1}{4} v_0 \]

### Step 3: Use the equation of motion
We can use the first equation of motion:
\[ v = u + at \]
where:
- \( v \) is the final velocity (\( \frac{1}{4} v_0 \)),
- \( u \) is the initial velocity (\( v_0 \)),
- \( a \) is the acceleration,
- \( t \) is the time (\( t_0 \)).

Substituting the known values:
\[ \frac{1}{4} v_0 = v_0 + a t_0 \]

### Step 4: Rearranging to find acceleration
Rearranging the equation gives:
\[ \frac{1}{4} v_0 - v_0 = a t_0 \]
\[ -\frac{3}{4} v_0 = a t_0 \]
Thus, we can express acceleration \( a \) as:
\[ a = -\frac{3 v_0}{4 t_0} \]

### Step 5: Determine the time until the body comes to rest
Now, we need to find the total time \( T \) until the velocity becomes zero. The body has a velocity of \( \frac{1}{4} v_0 \) at time \( t_0 \). Let \( t_2 \) be the additional time taken to come to rest from \( \frac{1}{4} v_0 \).

Using the same equation of motion:
\[ 0 = \frac{1}{4} v_0 + a t_2 \]
Substituting \( a \):
\[ 0 = \frac{1}{4} v_0 - \frac{3 v_0}{4 t_0} t_2 \]

### Step 6: Solve for \( t_2 \)
Rearranging gives:
\[ \frac{3 v_0}{4 t_0} t_2 = \frac{1}{4} v_0 \]
Dividing both sides by \( v_0 \) (assuming \( v_0 \neq 0 \)):
\[ \frac{3}{4 t_0} t_2 = \frac{1}{4} \]
Multiplying both sides by \( 4 t_0 \):
\[ 3 t_2 = t_0 \]
Thus, we find:
\[ t_2 = \frac{t_0}{3} \]

### Step 7: Calculate the total time of motion
The total time \( T \) until the body comes to rest is:
\[ T = t_0 + t_2 = t_0 + \frac{t_0}{3} = \frac{3 t_0}{3} + \frac{t_0}{3} = \frac{4 t_0}{3} \]

### Final Answer
The total time of motion of the body until its velocity becomes zero is:
\[ \boxed{\frac{4 t_0}{3}} \]