Area under a velocity-time graph gives

Options

A. the time taken by a moving object
B. the distance travelled by a moving object
C. the acceleration of moving object
D. the retardation of a moving object

Answer

To solve the question regarding what the area under a velocity-time graph represents, we can follow these steps:

### Step 1: Understand the Definitions
- **Velocity (v)** is defined as the rate of change of displacement with respect to time. Mathematically, this can be expressed as:
\[ v = \frac{ds}{dt} \]
where \( ds \) is the change in displacement and \( dt \) is the change in time.

### Step 2: Rearranging the Equation
- By rearranging the above equation, we can express \( ds \) in terms of \( v \) and \( dt \):
\[ ds = v \cdot dt \]

### Step 3: Integrating Over Time
- To find the total displacement over a time interval, we can integrate both sides:
\[ \int ds = \int v \, dt \]
This means that the total displacement \( s \) is equal to the integral of velocity over time.

### Step 4: Area Under the Curve
- The integral of velocity \( v \) with respect to time \( t \) gives us the area under the velocity-time graph. Therefore:
\[ s = \int v \, dt \]
This indicates that the area under the velocity-time graph corresponds to the total displacement.

### Step 5: Conclusion
- Thus, the area under a velocity-time graph gives us the **displacement** of the object over the time interval considered. If the motion is in a straight line and the velocity is constant, this displacement will also equal the distance traveled.

### Final Answer
The area under a velocity-time graph gives **displacement**.

---