### Step 1: Find the momentum vectors for each jogger **Jogger 1:** - Mass (\(m_1\)) = 85 kg - Velocity (\(v_1\)) = 2. m/s (due east) Momentum (\(p_1\)) = \(m_1 \times v_1 = 85 \times 2. = 170\) kg·m/s (due east) So, in component form: - \(p1x} = 170\) kg·m/s (east) - \(p_{1y} = \) kg·m/s (north) **Jogger 2:** - Mass (\(m_2\)) = 55 kg - Velocity (\(v_2\)) = 3. m/s (32° north of east) Momentum (\(p_2\)) = \(m_2 \times v_2 = 55 \times 3. = 165\) kg·m/s Components: - \(p_{2x} = 165 \cos(32^\circ) \approx 140\) kg·m/s - \(p_{2y} = 165 \sin(32^\circ) \approx 87\) kg·m/s **Explanation:** We broke down each jogger's momentum into x (east) and y (north) components for easier addition. --- ### Step 2: Add the momentum components and find magnitude and direction **Total x-component:** - \(p_{x, \text{total}} = p_{1x} + p_{2x} = 170 + 140 = 310\) kg·m/s **Total y-component:** - \(p_{y, \text{total}} = p_{1y} + p_{2y} = + 87 = 87\) kg·m/s **Magnitude of total momentum:** \[ p_{\text{total}} = \sqrt{(310)^2 + (87)^2} \approx \sqrt{96100 + 7569} \approx \sqrt{103669} \approx 322\ \text{kg·m/s} \] **Direction (north of east):** \[ \theta = \tan^{-1}\left(\frac{87}{310}\right) \approx \tan^{-1}(.281) \approx 15.7^\circ \] **Explanation:** We added the x and y components to get the total momentum, then used the Pythagorean theorem and inverse tangent to find the magnitude and direction. --- ### **Final Answer** - **Magnitude:** \(322\) kg·m/s - **Direction:** \(15.7^\circ\) north of east