Area under power-time curve gives total work done

Options

A. by the body only
B. on the body only
C. Both (a) and (b)
D. neither (a) nor (b)

Explanation

To solve the question "Area under power-time curve gives total work done," we will analyze the relationship between power, work, and the area under the curve in a step-by-step manner.

### Step-by-Step Solution:

1. **Understanding Power and Work**:
- Power (P) is defined as the rate at which work (W) is done. Mathematically, this is expressed as:
\[ P = \frac{dW}{dt} \]
- This means that the instantaneous power is the derivative of work with respect to time.

2. **Relating Power to Work**:
- Rearranging the equation gives us:
\[ dW = P \cdot dt \]
- This indicates that an infinitesimal amount of work done (dW) over an infinitesimal time interval (dt) is equal to the power multiplied by that time interval.

3. **Integrating to Find Total Work**:
- To find the total work done over a time interval, we integrate the power with respect to time:
\[ W = \int P \, dt \]
- The result of this integration gives us the total work done over the specified time interval.

4. **Interpreting the Area Under the Curve**:
- In a power-time graph, the area under the curve represents the integral of power with respect to time. Thus:
\[ \text{Area under the power-time curve} = W \]
- Therefore, the area under the power-time curve indeed gives us the total work done.

5. **Determining the Type of Work Done**:
- The question further asks whether the work done is "by the body" or "on the body."
- If the area under the curve is positive, it indicates that work is done **on the body** (meaning energy is supplied to the body).
- If the area is negative, it indicates that work is done **by the body** (meaning the body is doing work against an external force).

6. **Conclusion**:
- Since the area under the power-time curve can be either positive or negative depending on the situation, the correct answer to the question is that the area can represent both types of work depending on its sign.
- Therefore, the answer is option **3: both a and b**.