In which of the following graphs, the total change in momentum is zero ?

Options

A.
B.
C.
D.

Answer

To determine in which of the given graphs the total change in momentum is zero, we can follow these steps:

### Step 1: Understand the relationship between force and momentum
The change in momentum (Δp) is related to the force (F) applied over time (t) by the equation:
\[ F = \frac{dp}{dt} \]
This means that the force is equal to the rate of change of momentum with respect to time.

### Step 2: Integrate the force over time
To find the total change in momentum, we need to integrate the force with respect to time:
\[ \Delta p = \int F \, dt \]
The result of this integration gives us the area under the force-time graph.

### Step 3: Analyze the area under the graph
- If the area under the force-time graph is positive, it indicates a positive change in momentum.
- If the area is negative, it indicates a negative change in momentum.
- If the total area (considering positive and negative areas) is zero, then the total change in momentum is also zero.

### Step 4: Evaluate the given graphs
We need to look at each graph and calculate the areas:
1. **Graph A**: If the area is entirely above the x-axis, the total change in momentum is positive.
2. **Graph B**: If the area is also entirely above the x-axis, the total change in momentum is positive.
3. **Graph C**: If there are equal areas above and below the x-axis, the total area sums to zero, indicating that the total change in momentum is zero.
4. **Graph D**: If the area is entirely below the x-axis, the total change in momentum is negative.

### Step 5: Conclusion
From the analysis, the graph where the total change in momentum is zero is **Graph C**, as it has equal positive and negative areas that cancel each other out.

### Final Answer
The total change in momentum is zero in **Graph C**.

---