An object is subjected to retardation, `(dv)/(dt)=-5sqrtv` , which has initial velocity of `4ms^(-1)`. The time taken by the object to come to rest would be

Options

A. 2 s
B. `5/4 s`
C. `4/5 s`
D. `1/2 s`

Answer

To solve the problem step by step, we start with the given equation of motion:

1. **Write the equation of retardation:**
\[ \frac{dv}{dt} = -5\sqrt{v} \]

2. **Rearrange the equation:**
We can separate the variables by rearranging the equation:
\[ \frac{dv}{\sqrt{v}} = -5 dt \]

3. **Integrate both sides:**
We will integrate the left side from the initial velocity \( v = 4 \, \text{m/s} \) to the final velocity \( v = 0 \, \text{m/s} \), and the right side from \( t = 0 \) to \( t \):
\[ \int_{4}^{0} \frac{dv}{\sqrt{v}} = \int_{0}^{t} -5 dt \]

4. **Calculate the left side integral:**
The integral \( \int \frac{dv}{\sqrt{v}} \) is equal to \( 2\sqrt{v} \):
\[ \left[ 2\sqrt{v} \right]_{4}^{0} = 2\sqrt{0} - 2\sqrt{4} = 0 - 4 = -4 \]

5. **Calculate the right side integral:**
The integral \( \int -5 dt \) is equal to \( -5t \):
\[ \left[-5t\right]_{0}^{t} = -5t - 0 = -5t \]

6. **Set the integrals equal to each other:**
Now we equate the results from both integrals:
\[ -4 = -5t \]

7. **Solve for \( t \):**
Rearranging gives:
\[ 5t = 4 \implies t = \frac{4}{5} \text{ seconds} \]

Thus, the time taken by the object to come to rest is:
\[ t = \frac{4}{5} \text{ seconds} \]