### Step 1: Introduction / Overview The problem involves a piecewise constant acceleration-time graph. We are given the initial velocity and asked to determine the velocity at a specific time, as well as the average acceleration over a certain interval. To solve these, we will use concepts from kinematics: integrating acceleration to get change in velocity, and using the definition of average acceleration. --- ### Step 2: Finding Velocity at \( t = 15.25\,s \) #### **a) Reading Acceleration Intervals** From the graph: - \( 0 < t < 5s \): \( a = -5\,m/s^2 \) - \( 5 < t < 10s \): \( a = 0\,m/s^2 \) - \( 10 < t < 15s \): \( a = +5\,m/s^2 \) - \( 15 < t < 20s \): \( a = 0\,m/s^2 \) - \( 20 < t < 25s \): \( a = -5\,m/s^2 \) We want the velocity at \( t = 15.25\,s \). We need to calculate the change in velocity in each interval up to \( 15.25\,s \). #### **b) Calculation of Change in Velocity (\( \Delta v \))** Formula: \[ \Delta v = a \cdot \Delta t \] Make a table for each interval: | Interval | a (m/s²) | Δt (s) | Δv (m/s) = a × Δt | |-----------------|----------|----------|----------------------| | 0 – 5 s | -5 | 5 | -5 × 5 = -25 | | 5 – 10 s | 0 | 5 | 0 × 5 = 0 | | 10 – 15 s | +5 | 5 | 5 × 5 = 25 | | 15 – 15.25 s | 0 | 0.25 | 0 × 0.25 = 0 | #### **c) Total Change in Velocity** Add all \( \Delta v \) up to \( 15.25s \): \[ \Delta v_{total} = -25 + 0 + 25 + 0 = 0 \] #### **d) Find Final Velocity** \[ v_{15.25s} = v_0 + \Delta v_{total} \] Given: \( v_0 = -25\,m/s \) \[ v_{15.25s} = -25 + 0 = -25\,m/s \] --- ### Step 3: Calculating Average Acceleration from 3.25s to 22.25s #### **a) Find Total Change in Velocity** Breakdown the time intervals and corresponding accelerations: - 3.25 – 5.00 s: \( a = -5\,m/s^2, \Delta t = 1.75\,s \) - 5.00 – 10.00 s: \( a = 0\,m/s^2, \Delta t = 5\,s \) - 10.00 – 15.00 s: \( a = +5\,m/s^2, \Delta t = 5\,s \) - 15.00 – 20.00 s: \( a = 0\,m/s^2, \Delta t = 5\,s \) - 20.00 – 22.25 s: \( a = -5\,m/s^2, \Delta t = 2.25\,s \) Fill a table: | Interval | a (m/s²) | Δt (s) | Δv (m/s) | |-----------------|----------|---------|------------| | 3.25 – 5.00 s | -5 | 1.75 | -5 × 1.75 = -8.75 | | 5.00 – 10.00 s | 0 | 5 | 0 × 5 = 0 | | 10.00 – 15.00 s | +5 | 5 | 5 × 5 = 25 | | 15.00 – 20.00 s | 0 | 5 | 0 × 5 = 0 | | 20.00 – 22.25 s | -5 | 2.25 | -5 × 2.25 = -11.25 | Sum up \( \Delta v \): \[ \Delta v_{total} = -8.75 + 0 + 25 + 0 - 11.25 = 5\,m/s \] #### **b) Find Total Time Interval** \[ \Delta t_{total} = 22.25 - 3.25 = 19.00\,s \] #### **c) Calculate Average Acceleration** \[ a_{avg} = \frac{\Delta v_{total}}{\Delta t_{total}} = \frac{5}{19.00} = 0.263\,m/s^2 \] --- ### Final Answers **1. Velocity at time 15.25 s:** \[ \boxed{-25\,m/s} \] **2. Average acceleration from 3.25 s to 22.25 s:** \[ \boxed{0.263\,m/s^2} \]