https://file.aiquickdraw.com/mathbot/file/1725899837866dt2smzq1.jpg

The question in the image asks you to choose the option that correctly fills in the blanks in this sentence:

"The ______ velocity is the ______ of the tangent line on a ______ vs. time graph."

To break this down:

- "______ velocity" refers to either instantaneous or average velocity.
- The "______ of the tangent line" relates to the slope or integral of the line.
- The "______ vs. time" graph likely refers to a position vs. time or an acceleration vs. time graph.

### Let's consider the choices:
1. **A. instantaneous; slope; position**  
   - This suggests instantaneous velocity is the slope of the tangent line on a position vs. time graph.  
   - **This is correct** because the slope of a position vs. time graph gives the velocity, and "instantaneous" refers to the velocity at a specific point in time.
   
2. **B. average; integral; position**
   - This suggests average velocity is the integral of a position vs. time graph, which is incorrect. The integral of a velocity vs. time graph would give position, but this is asking about velocity.

3. **C. instantaneous; integral; position**
   - This suggests that instantaneous velocity is the integral of a position vs. time graph, which is incorrect. As noted earlier, the slope, not the integral, determines instantaneous velocity.

4. **D. average; slope; acceleration**
   - This option suggests that average velocity is the slope of an acceleration vs. time graph, which is incorrect. Velocity relates to the slope of a position vs. time graph, not acceleration.

### Conclusion:
The correct answer is **A. instantaneous; slope; position**.

Let me know if you would like more details or have further questions!

Here are five related questions you could explore:
1. What is the difference between instantaneous and average velocity?
2. How is the slope of a position vs. time graph related to velocity?
3. How can you calculate the velocity from a graph that shows acceleration vs. time?
4. What is the role of integrals in determining velocity or position from a graph?
5. How do the concepts of position, velocity, and acceleration relate to each other in kinematics?

**Tip:** The slope of a graph can tell you a lot about the physical meaning of the motion; always think about the units when interpreting it (e.g., meters per second for velocity).

Choose the option that identifies the correct terms that fill in the blanks: 'The ______ velocity is the ______ of the tangent line on a ______ vs. time graph.'

Learn how to identify instantaneous velocity as the slope of the tangent line on a position vs. time graph. This problem covers core concepts in kinematics and is suitable for high school physics students.

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