The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation
s = 4 sin(𝜋t) + 3 cos(𝜋t),
 where t is measured in seconds. (Round your answers to two decimal places.)
(a)
Find the average velocity during each time period.
(i)
[1, 2]
  
[1, 1.01]

 cm/s
(iv)
[1, 1.001]
  


To find the average velocity of the particle during the given time periods, we need to use the formula for average velocity:

\[
\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]

where \( s(t) \) is the displacement at time \( t \), and \( [t_1, t_2] \) is the time interval.

The displacement function is given as:

\[
s(t) = 4\sin(\pi t) + 3\cos(\pi t)
\]

Let's calculate the average velocity for each of the specified intervals.

### (i) Interval \([1, 2]\)

1. **Calculate \( s(1) \) and \( s(2) \)**:
   \[
   s(1) = 4\sin(\pi \cdot 1) + 3\cos(\pi \cdot 1) = 4\sin(\pi) + 3\cos(\pi) = 4(0) + 3(-1) = -3
   \]
   \[
   s(2) = 4\sin(\pi \cdot 2) + 3\cos(\pi \cdot 2) = 4\sin(2\pi) + 3\cos(2\pi) = 4(0) + 3(1) = 3
   \]

2. **Calculate the average velocity**:
   \[
   \text{Average velocity} = \frac{s(2) - s(1)}{2 - 1} = \frac{3 - (-3)}{1} = \frac{6}{1} = 6 \text{ cm/s}
   \]

### (ii) Interval \([1, 1.01]\)

1. **Calculate \( s(1.01) \)**:
   \[
   s(1.01) = 4\sin(\pi \cdot 1.01) + 3\cos(\pi \cdot 1.01)
   \]
   Using the approximations:
   \[
   \sin(\pi \cdot 1.01) \approx \sin(\pi + 0.01\pi) = -\sin(0.01\pi) \approx -0.0314
   \]
   \[
   \cos(\pi \cdot 1.01) \approx \cos(\pi + 0.01\pi) = -\cos(0.01\pi) \approx -0.9995
   \]
   \[
   s(1.01) \approx 4(-0.0314) + 3(-0.9995) = -0.1256 - 2.9985 \approx -3.1241
   \]

2. **Calculate the average velocity**:
   \[
   \text{Average velocity} = \frac{s(1.01) - s(1)}{1.01 - 1} = \frac{-3.1241 - (-3)}{0.01} = \frac{-0.1241}{0.01} = -12.41 \text{ cm/s}
   \]

### (iii) Interval \([1, 1.001]\)

1. **Calculate \( s(1.001) \)**:
   \[
   s(1.001) = 4\sin(\pi \cdot 1.001) + 3\cos(\pi \cdot 1.001)
   \]
   Using the approximations:
   \[
   \sin(\pi \cdot 1.001) \approx \sin(\pi + 0.001\pi) = -\sin(0.001\pi) \approx -0.00314
   \]
   \[
   \cos(\pi \cdot 1.001) \approx \cos(\pi + 0.001\pi) = -\cos(0.001\pi) \approx -0.999995
   \]
   \[
   s(1.001) \approx 4(-0.00314) + 3(-0.999995) = -0.01256 - 2.999985 \approx -3.012545
   \]

2. **Calculate the average velocity**:
   \[
   \text{Average velocity} = \frac{s(1.001) - s(1)}{1.001 - 1} = \frac{-3.012545 - (-3)}{0.001} = \frac{-0.012545}{0.001} = -12.55 \text{ cm/s}
   \]

### Final Results:

- **(i) Interval \([1, 2]\)**: Average velocity = \(6\) cm/s
- **(ii) Interval \([1, 1.01]\)**: Average velocity = \(-12.41\) cm/s
- **(iii) Interval \([1, 1.001]\)**: Average velocity = \(-12.55\) cm/s

Would you like any further details or explanations?

---

**Related Questions:**

1. What is the instantaneous velocity at \( t = 1 \) seconds?
2. How does the average velocity change as the time interval decreases?
3. Can you determine the acceleration of the particle at \( t = 1 \) second?
4. What is the maximum displacement of the particle and at what time does it occur?
5. How would the displacement function change if the sine and cosine coefficients were swapped?

**Tip:** Average velocity over very short intervals can be used to approximate instantaneous velocity.

The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation s = 4 sin(𝜋t) + 3 cos(𝜋t), where t is measured in seconds. (Round your answers to two decimal places.) (a) Find the average velocity during each time period. (i) [1, 2] [1, 1.01] (iv) [1, 1.001] cm/s

Calculate the average velocity of a particle moving along a straight line using the displacement function s = 4 sin(πt) + 3 cos(πt) for time intervals [1, 2], [1, 1.01], and [1, 1.001]. Learn key concepts in trigonometry and kinematics. Suitable for high school students in Grades 10-12.

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