The area of the acceleration-displacement curve of a body gives

Options

A. impulse
B. change in momentum per unit mass
C. change in KE per unit mass
D. total change is energy

Explanation

To solve the question regarding the area of the acceleration-displacement curve of a body, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
- We start with the relationship between acceleration (A), velocity (v), and displacement (x). Acceleration can be defined as the change in velocity over time:
\[ A = \frac{dv}{dt} \]
- We can also express this in terms of displacement:
\[ A = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v \]

2. **Integrating the Acceleration**:
- To find the area under the acceleration-displacement curve, we need to integrate the acceleration with respect to displacement. We can set up the integral from the initial velocity \( u \) to the final velocity \( v \):
\[ \text{Area} = \int_{u}^{v} A \, dx \]

3. **Using the Kinematic Equation**:
- From kinematics, we know that:
\[ v^2 = u^2 + 2a s \]
- Rearranging gives us:
\[ a = \frac{v^2 - u^2}{2s} \]
- Here, \( s \) is the displacement.

4. **Calculating the Change in Kinetic Energy**:
- The change in kinetic energy (KE) of the body as it moves from an initial speed \( u \) to a final speed \( v \) is given by:
\[ \Delta KE = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \]
- This can be expressed as:
\[ \Delta KE = \frac{m}{2} (v^2 - u^2) \]

5. **Finding Change in Kinetic Energy per Unit Mass**:
- To find the change in kinetic energy per unit mass, we divide the change in kinetic energy by the mass \( m \):
\[ \frac{\Delta KE}{m} = \frac{1}{2} (v^2 - u^2) \]

6. **Conclusion**:
- Thus, the area under the acceleration-displacement curve represents the change in kinetic energy per unit mass of the body. Therefore, the answer to the question is:
\[ \text{Area of acceleration-displacement curve} = \text{Change in kinetic energy per unit mass} \]

### Final Answer:
The area of the acceleration-displacement curve of a body gives the change in kinetic energy per unit mass.

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