The velocity v of a particle at any instant t moving in a straight line is given by v=s +1 where s meter is the distance travelled in t second what is the time taken by the particle to cover a distacne of 9m ?

Options

A. 1s
B. (log 10)s
C. 2 (log 10)s
D. 10 s

Explanation

To solve the problem step by step, we will start from the given information about the velocity of the particle and derive the time taken to cover a distance of 9 meters.

### Step 1: Understand the relationship between velocity, distance, and time.
The velocity \( v \) of the particle is given by the equation:
\[ v = s + 1 \]
where \( s \) is the distance traveled in meters and \( t \) is the time in seconds.

### Step 2: Relate velocity to distance and time.
We know that velocity can also be expressed as the derivative of distance with respect to time:
\[ v = \frac{ds}{dt} \]
Thus, we can set up the equation:
\[ \frac{ds}{dt} = s + 1 \]

### Step 3: Rearrange the equation for integration.
We can rearrange the equation to separate variables:
\[ \frac{ds}{s + 1} = dt \]

### Step 4: Integrate both sides.
Now, we will integrate both sides. The left side integrates with respect to \( s \) and the right side with respect to \( t \):
\[ \int \frac{ds}{s + 1} = \int dt \]
The left side integrates to:
\[ \ln |s + 1| = t + C \]
where \( C \) is the constant of integration.

### Step 5: Solve for the constant of integration.
To find the constant \( C \), we need an initial condition. However, since we are only interested in the time taken to cover a distance of 9 meters, we can proceed without finding \( C \) explicitly.

### Step 6: Substitute the distance into the equation.
We need to find the time \( t \) when the distance \( s = 9 \) meters:
\[ \ln |9 + 1| = t + C \]
This simplifies to:
\[ \ln 10 = t + C \]

### Step 7: Determine the time taken.
Since we do not have an initial condition, we can express the time taken as:
\[ t = \ln 10 - C \]
However, for our purposes, we can assume \( C = 0 \) for simplicity, so:
\[ t = \ln 10 \]

### Final Answer:
The time taken by the particle to cover a distance of 9 meters is:
\[ t = \ln 10 \text{ seconds} \]