Question 1204891: Consider the motion described by the vx-t graph of Fig. E2. 26. (see this link--> https://i.postimg.cc/SsGt7snK/Fig-E2-26.png )
(a) Calculate the area under the graph between t=0 and t=6.0s.
(b) For the time interval t=0 to t=6.0s, what is the magnitude of the average velocity of the cat?
(c) Use constant-acceleration equations to calculate the distance the cat travels in this time interval. How does your result compare to the area you calculated in part (a)?
problem from Young and Freedman. University Physics with Modern Physics Fifteenth Edition.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! Certainly, I can help you analyze the motion described by the vx-t graph (Fig. E2.26) from Young and Freedman's University Physics textbook.
**(a) Area under the graph**
The area under the vx-t graph represents the total displacement of the cat during the specified time interval. Since the velocity (vx) is positive throughout the interval (0 to 6.0 seconds), the area corresponds to the total positive displacement.
**Calculating the area:**
1. **Identify the shape:** The graph appears to be a trapezoid with a slanted top and a horizontal bottom.
2. **Base lengths:** The left base (b1) is 6.0 seconds (given) and the right base (b2) can be determined from the graph by measuring the horizontal extent of the graph at t = 6.0 seconds. Let's denote this measured value as b2.
3. **Height (h):** The height (h) of the trapezoid is the constant positive velocity (vx) throughout the interval. You can measure this value from the graph or it might be provided in the problem statement. Let's denote this value as h.
**Area formula for a trapezoid:**
```
Area = (b1 + b2) * h / 2
```
**Applying the formula:**
```
Area = (6.0 seconds + b2) * h / 2
```
**(b) Magnitude of the average velocity**
The average velocity represents the constant velocity that would result in the same displacement as the actual motion over the given time interval.
**Calculating the average velocity:**
Since the area under the curve represents the total displacement, we can use the following relationship:
```
Average velocity = Total displacement / Time interval
```
We already established that the total displacement is equal to the area under the curve calculated in part (a).
**Average velocity = Area / Time interval**
```
Average velocity = [(6.0 seconds + b2) * h / 2] / 6.0 seconds
```
**(c) Distance traveled using constant-acceleration equations**
If the problem provides information about the constant acceleration (a) of the cat, we can use the following constant-acceleration kinematic equations to calculate the distance traveled:
1. **Final velocity (vf):** We can potentially find the final velocity (vf) at t = 6.0 seconds from the graph or it might be given in the problem statement.
2. **Initial velocity (vi):** The initial velocity (vi) at t = 0 seconds can be determined from the graph (usually zero if the cat starts from rest).
**Possible equations (depending on the information provided):**
* If vf and vi are known:
```
d = (vf^2 - vi^2) / (2 * a)
```
* If vf and a are known:
```
d = vf * t - (a * t^2) / 2
```
**Comparison with the area**
The distance traveled using the constant-acceleration equations should be equal to the total displacement calculated in part (a) if the motion is purely positive (constant positive velocity throughout the interval). This is because the area under the vx-t graph represents the total positive displacement for motions with constant positive or negative velocity.
**Note:**
* If the cat experiences periods of negative velocity (vx goes below the x-axis), the constant-acceleration equations might not directly provide the total displacement. You would need to consider both positive and negative areas under the curve for such cases.
* Make sure to use consistent units (meters, seconds) when measuring from the graph and performing calculations.
By analyzing the vx-t graph and using the appropriate formulas, you can determine the displacement, average velocity, and distance traveled by the cat in the given time interval. The comparison between the area and the distance from constant-acceleration equations helps verify if the motion involves only positive velocity.
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