A particle moving along straight line has velocity `v = mu s` where s is in the displacement . If `s = s_0` then which of the following graph best represent s versus t

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the relationship between the velocity \( v \) and the displacement \( s \) given by the equation \( v = \mu s \). We will derive the expression for displacement \( s \) as a function of time \( t \) and then identify the appropriate graph that represents this relationship.

### Step-by-Step Solution:

1. **Understand the given relationship**:
We are given that the velocity \( v \) of a particle is proportional to its displacement \( s \) such that:
\[ v = \mu s \]
where \( \mu \) is a constant.

2. **Relate velocity to displacement**:
We know that velocity \( v \) can also be expressed as the rate of change of displacement with respect to time:
\[ v = \frac{ds}{dt} \]
Therefore, we can equate the two expressions for velocity:
\[ \frac{ds}{dt} = \mu s \]

3. **Rearrange the equation**:
We can rearrange this equation to separate variables:
\[ \frac{ds}{s} = \mu dt \]

4. **Integrate both sides**:
Now we will integrate both sides. The left side will be integrated with respect to \( s \) and the right side with respect to \( t \):
\[ \int \frac{ds}{s} = \int \mu dt \]
This gives us:
\[ \ln |s| = \mu t + C \]
where \( C \) is the constant of integration.

5. **Solve for the constant**:
To find the constant \( C \), we use the initial condition. At \( t = 0 \), the displacement \( s = s_0 \):
\[ \ln |s_0| = C \]

6. **Substitute back to find \( s \)**:
Substituting \( C \) back into the equation, we have:
\[ \ln |s| = \mu t + \ln |s_0| \]
Exponentiating both sides gives:
\[ |s| = e^{\mu t} \cdot |s_0| \]
Thus, we can express \( s \) as:
\[ s = s_0 e^{\mu t} \]

7. **Identify the graph**:
The equation \( s = s_0 e^{\mu t} \) represents an exponential growth function. At \( t = 0 \), \( s \) equals \( s_0 \), and as \( t \) increases, \( s \) increases exponentially. The graph of \( s \) versus \( t \) will be an upward-curving exponential graph starting from the point \( (0, s_0) \).

### Conclusion:
The graph that best represents \( s \) versus \( t \) is an exponential curve starting at \( s_0 \) when \( t = 0 \) and increasing as \( t \) increases.