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? $$ \vec{v}=2.2 \mathrm{~m} $$ $$ i $$

Question

UpStudy AI · Accepted Answer

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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