The acceleation fo a particle (a) is related to irs velocity (v) by ` a =- v`. What is the bature of velocity-time curve ?

Options

A. (a) Linearly increasing
B. (b) Exponentially decreasing
C. (c ) Exponentially increasing
D. (d) Linearly decreasing

Explanation

To solve the problem, we need to analyze the relationship between acceleration and velocity given by the equation \( a = -v \).

### Step-by-Step Solution:

1. **Understanding the Given Relation**:
We start with the equation given in the problem:
\[ a = -v \]
Here, \( a \) is the acceleration and \( v \) is the velocity of the particle.

2. **Expressing Acceleration in Terms of Velocity**:
We know that acceleration can also be expressed as the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Therefore, we can rewrite the equation as:
\[ \frac{dv}{dt} = -v \]

3. **Separating Variables**:
To solve this differential equation, we separate the variables \( v \) and \( t \):
\[ \frac{dv}{v} = -dt \]

4. **Integrating Both Sides**:
Now, we integrate both sides:
\[ \int \frac{dv}{v} = \int -dt \]
This gives us:
\[ \ln |v| = -t + C \]
where \( C \) is the constant of integration.

5. **Exponentiating to Solve for Velocity**:
To eliminate the natural logarithm, we exponentiate both sides:
\[ |v| = e^{-t + C} = e^C e^{-t} \]
Let \( e^C = k \) (a positive constant), we can write:
\[ v = k e^{-t} \]
where \( k \) is a constant that can be determined by initial conditions.

6. **Analyzing the Velocity-Time Relationship**:
The equation \( v = k e^{-t} \) shows that as time \( t \) increases, the velocity \( v \) decreases exponentially. This indicates that the velocity approaches zero as time goes to infinity.

7. **Determining the Nature of the Velocity-Time Curve**:
The velocity-time graph for this relationship is an exponential decay curve, starting from \( k \) (the initial velocity) at \( t = 0 \) and approaching zero as \( t \) increases.

### Conclusion:
The nature of the velocity-time curve is **exponentially decreasing**.