Gwen runs in a straight line with an average velocity of $$ \vec{v}_{1}=(6.8 \mathrm{~m} / \mathrm{s}) \hat{i} $$ for $$ t_{1}=7.7 \mathrm{~min} $$, and then with an average velocity of $$ \vec{v}_{2}=(-7.1 \mathrm{~m} / \mathrm{s}) \hat{i} $$ for $$ t_{2}=3.8 \mathrm{~min} $$. Providing the answers in the correct numbers of significant Figures: a) Find the displacement during $$ t_{1} $$ ? $$ \vec{d}_{1}=9 \quad \hat{i} $$ b) Find the overall displacement? $$ \vec{d}=\square \hat{i} $$ c) Find the overall average velocity? $$ \vec{v}=2.2 \mathrm{~m} $$ $$ i $$

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Let's solve each part of Gwen's motion problem step by step, ensuring that all answers are provided with the correct number of significant figures (2 significant figures in this case).

---

### Given:
- **First Phase:**
  - **Velocity ($$ \vec{v}_1 $$)**: $$ 6.8 \, \text{m/s} \, \hat{i} $$
  - **Time ($$ t_1 $$)**: $$ 7.7 \, \text{minutes} $$
  
- **Second Phase:**
  - **Velocity ($$ \vec{v}_2 $$)**: $$ -7.1 \, \text{m/s} \, \hat{i} $$
  - **Time ($$ t_2 $$)**: $$ 3.8 \, \text{minutes} $$

**Note:** All times will be converted to seconds since velocity is given in meters per second (m/s).

---

### **a) Displacement During $$ t_1 $$ ($$ \vec{d}_1 $$)**

**Steps:**
1. **Convert Time to Seconds:**
   $$
   t_1 = 7.7 \, \text{minutes} \times 60 \, \frac{\text{seconds}}{\text{minute}} = 462 \, \text{seconds}
   $$
   
2. **Calculate Displacement:**
   $$
   \vec{d}_1 = \vec{v}_1 \times t_1 = 6.8 \, \text{m/s} \times 462 \, \text{s} = 3141.6 \, \text{m} \, \hat{i}
   $$
   
3. **Round to 2 Significant Figures:**
   $$
   \vec{d}_1 \approx 3.1 \times 10^{3} \, \text{m} \, \hat{i}
   $$

**Answer:**
$$
\vec{d}_1 = 3.1 \times 10^{3} \, \text{m} \, \hat{i}
$$

---

### **b) Overall Displacement ($$ \vec{d} $$)**

**Steps:**
1. **Convert Second Phase Time to Seconds:**
   $$
   t_2 = 3.8 \, \text{minutes} \times 60 \, \frac{\text{seconds}}{\text{minute}} = 228 \, \text{seconds}
   $$
   
2. **Calculate Displacement for Second Phase ($$ \vec{d}_2 $$):**
   $$
   \vec{d}_2 = \vec{v}_2 \times t_2 = (-7.1 \, \text{m/s}) \times 228 \, \text{s} = -1618.8 \, \text{m} \, \hat{i}
   $$
   
3. **Calculate Total Displacement:**
   $$
   \vec{d} = \vec{d}_1 + \vec{d}_2 = 3141.6 \, \text{m} - 1618.8 \, \text{m} = 1522.8 \, \text{m} \, \hat{i}
   $$
   
4. **Round to 2 Significant Figures:**
   $$
   \vec{d} \approx 1.5 \times 10^{3} \, \text{m} \, \hat{i}
   $$

**Answer:**
$$
\vec{d} = 1.5 \times 10^{3} \, \text{m} \, \hat{i}
$$

---

### **c) Overall Average Velocity ($$ \vec{v}_{\text{avg}} $$)**

**Steps:**
1. **Calculate Total Time:**
   $$
   t_{\text{total}} = t_1 + t_2 = 462 \, \text{s} + 228 \, \text{s} = 690 \, \text{seconds}
   $$
   
2. **Calculate Average Velocity:**
   $$
   \vec{v}_{\text{avg}} = \frac{\vec{d}}{t_{\text{total}}} = \frac{1522.8 \, \text{m}}{690 \, \text{s}} \approx 2.206 \, \text{m/s} \, \hat{i}
   $$
   
3. **Round to 2 Significant Figures:**
   $$
   \vec{v}_{\text{avg}} \approx 2.2 \, \text{m/s} \, \hat{i}
   $$

**Answer:**
$$
\vec{v}_{\text{avg}} = 2.2 \, \text{m/s} \, \hat{i}
$$

---

### **Summary of Answers:**

a) **Displacement during $$ t_1 $$:**
$$
\vec{d}_1 = 3.1 \times 10^{3} \, \text{m} \, \hat{i}
$$

b) **Overall Displacement:**
$$
\vec{d} = 1.5 \times 10^{3} \, \text{m} \, \hat{i}
$$

c) **Overall Average Velocity:**
$$
\vec{v}_{\text{avg}} = 2.2 \, \text{m/s} \, \hat{i}
$$
a) Displacement during $$ t_1 $$: $$ \vec{d}_1 = 3.1 \times 10^{3} \, \text{m} \, \hat{i} $$ b) Overall displacement: $$ \vec{d} = 1.5 \times 10^{3} \, \text{m} \, \hat{i} $$ c) Overall average velocity: $$ \vec{v}_{\text{avg}} = 2.2 \, \text{m/s} \, \hat{i} $$

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