`u=(1)/(2)at^(2)`
The velocity,v,of an object t seconds after beginning to accelerate from rest at a constant acceleration,a, can be found using the equation above. According to the formula, what is the ratio of velocity of the object t seconds after begins to accelerate to the velocity of the object 2.5t seconds after the object begins to accelerate?

Options

A. `(4)/(25)`
B. `(2)/(5)`
C. `(5)/(2)`
D. `(25)/(4)`

Answer

To solve the problem, we need to find the ratio of the velocity of an object after \( t \) seconds to the velocity of the same object after \( 2.5t \) seconds, using the equation \( u = \frac{1}{2} a t^2 \).

### Step-by-Step Solution:

1. **Determine the velocity after \( t \) seconds**:
- Using the formula, we substitute \( t \) into the equation:
\[ u_t = \frac{1}{2} a t^2 \]

2. **Determine the velocity after \( 2.5t \) seconds**:
- Now, we substitute \( 2.5t \) into the same equation:
\[ u_{2.5t} = \frac{1}{2} a (2.5t)^2 \]
- Simplifying this, we calculate \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
- Therefore, we have:
\[ u_{2.5t} = \frac{1}{2} a (6.25t^2) = \frac{1}{2} a \cdot 6.25 t^2 \]

3. **Set up the ratio of the two velocities**:
- We now find the ratio \( \frac{u_t}{u_{2.5t}} \):
\[ \frac{u_t}{u_{2.5t}} = \frac{\frac{1}{2} a t^2}{\frac{1}{2} a \cdot 6.25 t^2} \]

4. **Simplify the ratio**:
- The \( \frac{1}{2} a \) and \( t^2 \) terms cancel out:
\[ \frac{u_t}{u_{2.5t}} = \frac{1}{6.25} \]

5. **Convert \( 6.25 \) to a fraction**:
- We know \( 6.25 = \frac{625}{100} \), hence:
\[ \frac{1}{6.25} = \frac{100}{625} \]

6. **Simplify \( \frac{100}{625} \)**:
- Dividing both the numerator and denominator by 25:
\[ \frac{100 \div 25}{625 \div 25} = \frac{4}{25} \]

### Final Answer:
The ratio of the velocity of the object after \( t \) seconds to the velocity of the object after \( 2.5t \) seconds is:
\[ \frac{4}{25} \]