Let's solve the questions step by step using the provided acceleration vs. time graph! ### **1. What is the velocity at time 15.25 s?** #### **Step 1: Identify Initial Conditions** - Initial velocity \( v_0 = -25 \) m/s at \( t_0 = 0 \) s #### **Step 2: Find the Change in Velocity** The change in velocity (\(\Delta v\)) is the area under the acceleration-time graph between \( t = 0 \) s and \( t = 15.25 \) s. Break it into segments: - **From \( 0 \) to \( 5 \) s:** \( a = -5 \) m/s² - **From \( 5 \) to \( 10 \) s:** \( a = 0 \) m/s² - **From \( 10 \) to \( 15 \) s:** \( a = +5 \) m/s² - **From \( 15 \) to \( 15.25 \) s:** \( a = 0 \) m/s² #### **Step 3: Calculate Area (Change in Velocity) in Each Segment** - **First segment \( (0 \rightarrow 5 \) s\() :** \[ \Delta v_1 = a \cdot \Delta t = (-5)(5 - 0) = -25 \text{ m/s} \] - **Second segment \( (5 \rightarrow 10 \) s\() :** \[ \Delta v_2 = (0)(10 - 5) = 0 \text{ m/s} \] - **Third segment \( (10 \rightarrow 15 \) s\() :** \[ \Delta v_3 = (5)(15 - 10) = 25 \text{ m/s} \] - **Fourth segment \( (15 \rightarrow 15.25 \) s\() :** \[ \Delta v_4 = (0)(15.25 - 15) = 0 \text{ m/s} \] #### **Step 4: Total Change in Velocity** \[ \Delta v_{\text{total}} = \Delta v_1 + \Delta v_2 + \Delta v_3 + \Delta v_4 = -25 + 0 + 25 + 0 = 0 \text{ m/s} \] #### **Step 5: Find Final Velocity** \[ v_{15.25} = v_0 + \Delta v_{\text{total}} = -25 + 0 = \boxed{-25~\text{m/s}} \] --- ### **2. What is the average acceleration during the interval from 3.25 s to 22.25 s?** #### **Step 1: Find Total Change in Velocity (\(\Delta v\))** Breakdown by intervals: - **From 3.25 to 5 s:** \( a = -5 \) m/s², duration = \(5 - 3.25 = 1.75\) s \[ \Delta v_A = (-5) \times 1.75 = -8.75 \text{ m/s} \] - **From 5 to 10 s:** \( a = 0 \), duration = 5 s \[ \Delta v_B = 0 \] - **From 10 to 15 s:** \( a = 5 \), duration = 5 s \[ \Delta v_C = 5 \times 5 = 25 \text{ m/s} \] - **From 15 to 20 s:** \( a = 0 \), duration = 5 s \[ \Delta v_D = 0 \] - **From 20 to 22.25 s:** \( a = -5 \), duration = \(22.25 - 20 = 2.25\) s \[ \Delta v_E = (-5) \times 2.25 = -11.25 \text{ m/s} \] Sum: \[ \Delta v_{\text{total}} = -8.75 + 0 + 25 + 0 + (-11.25) = 5.0 \text{ m/s} \] #### **Step 2: Find Time Interval** \[ \Delta t = 22.25 - 3.25 = 19.0~\text{s} \] #### **Step 3: Calculate Average Acceleration** \[ a_{\text{avg}} = \frac{\Delta v_{\text{total}}}{\Delta t} = \frac{5.0}{19.0} \approx \boxed{0.26~\text{m/s}^2} \] --- ## **Final Answers** 1. **Velocity at 15.25 s:**                                   \(\boxed{-25~\text{m/s}}\) 2. **Average acceleration (3.25 s to 22.25 s):**                         \(\boxed{0.26~\text{m/s}^2}\)