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 will follow these steps:

### Step 1: Understand the Variables
- The x-axis represents displacement (s).
- The y-axis represents acceleration (a).

### Step 2: Define the Area Under the Curve
- The area under the acceleration-displacement curve can be represented as the integral of acceleration with respect to displacement.

### Step 3: Set Up the Integral
- The small elemental area (dA) can be expressed as:
\[ dA = a \cdot ds \]
- To find the total area, we need to integrate this from an initial displacement \(s_1\) to a final displacement \(s_2\):
\[ A = \int_{s_1}^{s_2} a \, ds \]

### Step 4: Relate Acceleration to Velocity
- Recall that acceleration \(a\) is the rate of change of velocity \(v\):
\[ a = \frac{dv}{dt} \]
- We can express \(ds\) in terms of \(v\):
\[ ds = v \, dt \]
- Thus, we can rewrite the integral as:
\[ A = \int_{v_1}^{v_2} v \, dv \]

### Step 5: Perform the Integration
- The integral of \(v\) with respect to \(v\) is:
\[ A = \frac{1}{2} v^2 \bigg|_{v_1}^{v_2} = \frac{1}{2} (v_2^2 - v_1^2) \]

### Step 6: Relate to Kinetic Energy
- The change in kinetic energy (\(\Delta K\)) is given by:
\[ \Delta K = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 \]
- Therefore, the area under the curve can be expressed as:
\[ A = \frac{1}{2m} \Delta K \]

### Step 7: Conclusion
- The area under the acceleration-displacement curve represents the change in kinetic energy per unit mass:
\[ \text{Area under the curve} = \text{Change in kinetic energy per unit mass} \]

### 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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