Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. (Note: the acceleration due to gravity is $$ 9.8 \mathrm{~m} / \mathrm{s}^{2} $$.) a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point (What is the height?) d. Find the time when the object strikes the ground. A payload is released at an elevation of 500 m from a hot-air balloon that is rising at a rate of $$ 10 \mathrm{~m} / \mathrm{s} $$. a. $$ v(t)=10-9.8 t $$ b. $$ s(t)=500+10 t-4.9 t^{2} $$ c. The object's highest point is 505.10 m at time $$ \mathrm{t}=1.02 \mathrm{~s} $$. (Simplify your answers. Round to two decimal places as needed.) d. $$ \mathrm{t}=11.18 $$ (Simplify your answer. Round to two decimal places as needed.)

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Let's analyze the vertical motion of the payload released from the hot-air balloon step by step. We'll verify each part of your solution and provide detailed explanations.

**Given:**
- **Initial Elevation ($$ s_0 $$)**: 500 meters
- **Initial Velocity ($$ v_0 $$)**: 10 m/s (since the balloon is rising)
- **Acceleration due to Gravity ($$ a $$)**: $$ -9.8 \, \text{m/s}^2 $$ (negative since it's acting downward)

### a. Finding the Velocity as a Function of Time ($$ v(t) $$)

**Formula:**
The velocity of an object under constant acceleration is given by:
$$ v(t) = v_0 + a t $$

**Calculation:**
$$ v(t) = 10 \, \text{m/s} + (-9.8 \, \text{m/s}^2) \cdot t $$
$$ v(t) = 10 - 9.8t \, \text{m/s} $$

**Your Answer:** $$ v(t) = 10 - 9.8t $$  
**Verification:** **Correct**

---

### b. Finding the Position as a Function of Time ($$ s(t) $$)

**Formula:**
The position of an object under constant acceleration is given by:
$$ s(t) = s_0 + v_0 t + \frac{1}{2} a t^2 $$

**Calculation:**
$$ s(t) = 500 \, \text{m} + 10 \, \text{m/s} \cdot t + \frac{1}{2} \cdot (-9.8 \, \text{m/s}^2) \cdot t^2 $$
$$ s(t) = 500 + 10t - 4.9t^2 \, \text{m} $$

**Your Answer:** $$ s(t) = 500 + 10t - 4.9t^2 $$  
**Verification:** **Correct**

---

### c. Determining the Time and Height at the Highest Point

**Finding the Time ($$ t_{\text{max}} $$) When Velocity is Zero:**

At the highest point, the vertical velocity becomes zero.
$$ v(t_{\text{max}}) = 0 $$
$$ 0 = 10 - 9.8t_{\text{max}} $$
$$ 9.8t_{\text{max}} = 10 $$
$$ t_{\text{max}} = \frac{10}{9.8} \approx 1.02 \, \text{seconds} $$

**Finding the Height ($$ s_{\text{max}} $$) at $$ t_{\text{max}} $$:**
$$ s(t_{\text{max}}) = 500 + 10(1.02) - 4.9(1.02)^2 $$
$$ s(t_{\text{max}}) = 500 + 10.2 - 4.9(1.0404) $$
$$ s(t_{\text{max}}) = 500 + 10.2 - 5.10 $$
$$ s(t_{\text{max}}) = 505.10 \, \text{meters} $$

**Your Answers:**  
- **Time:** $$ t = 1.02 \, \text{s} $$  
- **Height:** $$ 505.10 \, \text{m} $$  
**Verification:** **Both Correct**

---

### d. Determining the Time When the Object Strikes the Ground

**Condition for Striking the Ground:**
$$ s(t) = 0 $$

**Setting Up the Equation:**
$$ 500 + 10t - 4.9t^2 = 0 $$

**Solving the Quadratic Equation:**
$$ -4.9t^2 + 10t + 500 = 0 $$
$$ 4.9t^2 - 10t - 500 = 0 $$

**Using the Quadratic Formula:**
$$ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
where $$ a = 4.9 $$, $$ b = -10 $$, and $$ c = -500 $$.

$$ t = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 4.9 \cdot (-500)}}{2 \cdot 4.9} $$
$$ t = \frac{10 \pm \sqrt{100 + 9800}}{9.8} $$
$$ t = \frac{10 \pm \sqrt{9900}}{9.8} $$
$$ \sqrt{9900} \approx 99.50 $$
$$ t = \frac{10 \pm 99.50}{9.8} $$

**Considering Only the Positive Root (since time cannot be negative):**
$$ t = \frac{10 + 99.50}{9.8} $$
$$ t = \frac{109.50}{9.8} $$
$$ t \approx 11.18 \, \text{seconds} $$

**Your Answer:** $$ t = 11.18 \, \text{s} $$  
**Verification:** **Correct**

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### **Summary of Results:**

a. **Velocity Function:** $$ v(t) = 10 - 9.8t \, \text{m/s} $$  
b. **Position Function:** $$ s(t) = 500 + 10t - 4.9t^2 \, \text{m} $$  
c. **Highest Point:**  
 - **Time:** $$ 1.02 \, \text{seconds} $$  
 - **Height:** $$ 505.10 \, \text{meters} $$  
d. **Time to Strike Ground:** $$ 11.18 \, \text{seconds} $$

**All your answers are correct! Well done!**
All your answers are correct! Well done!

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