The velocity $v$ of a particle at any in moving in a straight line is given by $v=s+1$where $s$ metre is the distance travelled in $t$seconds. What is the time taken by the particle to cover a distance of $9m$?

The correct option is B

$\log \left(10\right)s$

Explanation for the correct answer:-

In the case of option (B),

Step 1. Given Data,

$v=s+1$

$s=9m$

Step 2. Formula used,

Velocity can be written as $v=\frac{ds}{dt}$

$s$ is the distance traveled.

$ds$is the change in distance

$dt$ is the change in time

$t$ is the time taken.

Step 3. Calculate the time,

Putting the given data,

$\frac{ds}{dt}=s+1$

$\frac{\mathrm{ds}}{\mathrm{s}+1}=\mathrm{dt}$

Integrating both sides, we get as

$\int \frac{ds}{s+1}=\int dt$

$\log (s+1)=t$

At $s=9m$

$\log (9+1)=t$

$\log 10=t$

$t=\log \left(10\right)sec$

The time taken by the particle to cover a distance of $9m$ is $\log \left(10\right)s$

Explanation for the incorrect answer:-

Option (A) $1s$

1. We cannot state that velocity is constant if speed, or the amount of velocity, is constant but the direction of movement changes.
2. Option A is incorrect

Option (C) $2\log \left(10\right)s$

1. There is a predisposition to think that an object has no acceleration if it is travelling at a constant speed.
2. In the case of an object moving in a straight line, this is in fact false.
3. Option c is incorrect

Option (D) $10s$

1. On the other hand, whether or not the speed is changing, a particle traveling down a curved path accelerates.
2. The acceleration of a particle must be 0 if its speed is constant.
3. Constant speed does not mean there is no acceleration.
4. Hence it is incorrect.

Therefore, option (B) is the correct answer.