Check point 8 In Figure 1.48 , an 8001 g car travels at \( 20 \mathrm{~ms} s^{\prime} \) east. It colldes with a 3000 kg truck travelling at \( 12 \mathrm{~ms} s^{-1} \) wot. The collision lasts \( 0.5 s \). After the collision, the truck moves at \( 4 \mathrm{~ms} s^{1} \) west

To analyze the collision between the car and the truck, we can use the principles of conservation of momentum. Here are the known conditions extracted from the problem: 1. Mass of the car, \( m_c = 8001 \, \text{g} = 8.001 \, \text{kg} \) 2. Velocity of the car before the collision, \( v_{c_i} = 20 \, \text{m/s} \) (east) 3. Mass of the truck, \( m_t = 3000 \, \text{kg} \) 4. Velocity of the truck before the collision, \( v_{t_i} = -12 \, \text{m/s} \) (west, negative direction) 5. Duration of the collision, \( t = 0.5 \, \text{s} \) 6. Velocity of the truck after the collision, \( v_{t_f} = -4 \, \text{m/s} \) (west, negative direction) ### Step 1: Calculate the initial momentum of the system The total initial momentum \( p_i \) of the system (car + truck) can be calculated using the formula: \[ p_i = m_c \cdot v_{c_i} + m_t \cdot v_{t_i} \] ### Step 2: Calculate the final momentum of the system The total final momentum \( p_f \) of the system after the collision can be calculated as: \[ p_f = m_c \cdot v_{c_f} + m_t \cdot v_{t_f} \] Where \( v_{c_f} \) is the final velocity of the car after the collision, which we need to find. ### Step 3: Apply the conservation of momentum According to the conservation of momentum: \[ p_i = p_f \] ### Step 4: Solve for the final velocity of the car We can rearrange the equation to solve for \( v_{c_f} \): \[ m_c \cdot v_{c_i} + m_t \cdot v_{t_i} = m_c \cdot v_{c_f} + m_t \cdot v_{t_f} \] Now, let's calculate the initial momentum and set up the equation to find \( v_{c_f} \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(8001\times 20+3000\left(-12\right)\) - step1: Multiply the numbers: \(160020+3000\left(-12\right)\) - step2: Multiply the numbers: \(160020-36000\) - step3: Subtract the numbers: \(124020\) Solve the equation \( 8001 * 20 + 3000 * (-12) = 8001 * v_c_f + 3000 * (-4) \). Solve the equation by following steps: - step0: Solve for \(v_{c}\): \(8001\times 20+3000\left(-12\right)=8001v_{c}+3000\left(-4\right)\) - step1: Simplify: \(124020=8001v_{c}-12000\) - step2: Swap the sides: \(8001v_{c}-12000=124020\) - step3: Move the constant to the right side: \(8001v_{c}=124020+12000\) - step4: Add the numbers: \(8001v_{c}=136020\) - step5: Divide both sides: \(\frac{8001v_{c}}{8001}=\frac{136020}{8001}\) - step6: Divide the numbers: \(v_{c}=\frac{45340}{2667}\) Calculate or simplify the expression \( 45340/2667 \). Calculate the value by following steps: - step0: Calculate: \(\frac{45340}{2667}\) ### Step 1: Calculate the initial momentum The initial momentum \( p_i \) of the system is calculated as follows: \[ p_i = 8001 \cdot 20 + 3000 \cdot (-12) = 124020 \, \text{kg m/s} \] ### Step 2: Set up the equation for final momentum Using the conservation of momentum, we set up the equation: \[ 124020 = 8001 \cdot v_{c_f} + 3000 \cdot (-4) \] ### Step 3: Solve for the final velocity of the car Rearranging the equation gives: \[ 8001 \cdot v_{c_f} = 124020 + 12000 \] \[ 8001 \cdot v_{c_f} = 136020 \] Now, solving for \( v_{c_f} \): \[ v_{c_f} = \frac{136020}{8001} \] Calculating this gives: \[ v_{c_f} \approx 17.0 \, \text{m/s} \] ### Conclusion The final velocity of the car after the collision is approximately \( 17.0 \, \text{m/s} \) east.