The acceleration (a) of a particle depends on displacement (s) as `a = 5 + s` . Initially `s = 0, v = 5`, then velocity `v` corresponding to the displacement is given by

Options

A. A) `v = 5 + s`
B. B) `v = sqrt(5 + s)`
C. C) `v = sqrt(s^2 + 10s)`
D. D) `v = s - 5`

Answer

To solve the problem, we need to find the velocity \( v \) as a function of displacement \( s \) given the relationship between acceleration \( a \) and displacement \( s \).

### Step-by-Step Solution:

1. **Write the given equation for acceleration**:
\[ a = 5 + s \]

2. **Relate acceleration to velocity**:
We know that acceleration \( a \) can be expressed as:
\[ a = \frac{dv}{dt} \]
But we can also express it in terms of displacement \( s \):
\[ a = v \frac{dv}{ds} \]
Hence, we can write:
\[ v \frac{dv}{ds} = 5 + s \]

3. **Separate the variables**:
Rearranging the equation gives:
\[ v \, dv = (5 + s) \, ds \]

4. **Integrate both sides**:
Now we integrate both sides:
\[ \int v \, dv = \int (5 + s) \, ds \]
The left side integrates to:
\[ \frac{v^2}{2} \]
The right side integrates to:
\[ 5s + \frac{s^2}{2} + C \]
Thus, we have:
\[ \frac{v^2}{2} = 5s + \frac{s^2}{2} + C \]

5. **Apply initial conditions**:
We know that initially when \( s = 0 \), \( v = 5 \). Plugging these values into the equation:
\[ \frac{5^2}{2} = 5(0) + \frac{0^2}{2} + C \]
This simplifies to:
\[ \frac{25}{2} = C \]

6. **Substitute \( C \) back into the equation**:
Now substituting \( C \) back into the equation gives:
\[ \frac{v^2}{2} = 5s + \frac{s^2}{2} + \frac{25}{2} \]

7. **Multiply through by 2**:
To eliminate the fraction, multiply the entire equation by 2:
\[ v^2 = 10s + s^2 + 25 \]

8. **Rearrange the equation**:
Rearranging gives:
\[ v^2 = s^2 + 10s + 25 \]

9. **Factor the right-hand side**:
The right-hand side can be factored as:
\[ v^2 = (s + 5)^2 \]

10. **Take the square root**:
Taking the square root of both sides gives:
\[ v = s + 5 \]

### Final Answer:
Thus, the velocity \( v \) corresponding to the displacement \( s \) is:
\[ v = s + 5 \]