The slope of kinetic energy displacement curve of a particle in motion is

Options

A. equal to the acceleration of the particle
B. directly proportional to the acceleration of the particle
C. inversely proportional to the acceleration of the particle
D. None of the above

Explanation

To solve the question regarding the slope of the kinetic energy displacement curve of a particle in motion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Kinetic Energy Formula**:
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

2. **Relate Velocity to Displacement**:
Velocity can be expressed as the derivative of displacement with respect to time:
\[ v = \frac{dx}{dt} \]

3. **Differentiate Kinetic Energy with Respect to Displacement**:
To find the slope of the kinetic energy displacement curve, we need to differentiate the kinetic energy with respect to displacement \( x \):
\[ \frac{d(KE)}{dx} = \frac{d}{dx} \left( \frac{1}{2} mv^2 \right) \]

4. **Apply the Chain Rule**:
Using the chain rule, we can differentiate \( v^2 \) with respect to \( x \):
\[ \frac{d(KE)}{dx} = \frac{1}{2} m \cdot 2v \cdot \frac{dv}{dx} = mv \frac{dv}{dx} \]

5. **Relate \( \frac{dv}{dx} \) to Acceleration**:
We know that acceleration \( a \) is defined as:
\[ a = \frac{dv}{dt} \]
and from the chain rule, we can express \( \frac{dv}{dx} \) as:
\[ \frac{dv}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx} = a \cdot \frac{1}{v} \]

6. **Substitute Back into the Equation**:
Now substitute \( \frac{dv}{dx} \) back into the equation for the slope:
\[ \frac{d(KE)}{dx} = mv \left(a \cdot \frac{1}{v}\right) = ma \]

7. **Conclusion**:
The slope of the kinetic energy displacement curve is:
\[ \frac{d(KE)}{dx} = ma \]
This shows that the slope is directly proportional to the acceleration \( a \) of the particle.

### Final Answer:
The slope of the kinetic energy displacement curve of a particle in motion is **directly proportional to the acceleration of the particle**.