The body moves along the `x-`axis . At time `t=0` the velocity of particle is `+0.5m//s`. The acceleration of particle in different time interval is given as`:`
`a={{:(-1 m//s^(2)," ",0slt=tlt=3s),(-2 m//s^(2)," ",3s lt tlt6s),(+4m//s^(2)," ",6s lt=tlt=10s):}`

Options

A. The speed of particle at `t=6 s` is zero.
B. The speed of particle at `t=6s` is `6m//s`.
C. The speed of particle of at `t=10 s` is `25 m//s`
D. The speed of particle at `t=10s` is `12m//s`.

Explanation

To solve the problem, we will calculate the velocity of the particle at different time intervals using the given acceleration values.

### Step-by-Step Solution:

1. **Initial Conditions**:
- At time \( t = 0 \), the initial velocity \( u = 0.5 \, \text{m/s} \).

2. **Acceleration from \( t = 0 \) to \( t = 3 \) seconds**:
- Given acceleration \( a = -1 \, \text{m/s}^2 \).
- Using the formula \( v = u + at \):
\[ v(3) = 0.5 + (-1) \times 3 \]
\[ v(3) = 0.5 - 3 = -2.5 \, \text{m/s} \]

3. **Acceleration from \( t = 3 \) to \( t = 6 \) seconds**:
- Now, the new initial velocity \( u = -2.5 \, \text{m/s} \) and acceleration \( a = -2 \, \text{m/s}^2 \).
- Again using the formula \( v = u + at \):
\[ v(6) = -2.5 + (-2) \times 3 \]
\[ v(6) = -2.5 - 6 = -8.5 \, \text{m/s} \]

4. **Acceleration from \( t = 6 \) to \( t = 10 \) seconds**:
- Now, the new initial velocity \( u = -8.5 \, \text{m/s} \) and acceleration \( a = +4 \, \text{m/s}^2 \).
- Using the formula \( v = u + at \):
\[ v(10) = -8.5 + 4 \times 4 \]
\[ v(10) = -8.5 + 16 = 7.5 \, \text{m/s} \]

5. **Final Result**:
- The velocity of the particle at \( t = 10 \) seconds is \( 7.5 \, \text{m/s} \).

### Summary of Results:
- The calculated velocity of the particle at \( t = 10 \) seconds is \( 7.5 \, \text{m/s} \).