Let's break the problem into steps: ## **Given Data** - Both cars travel at initial speed, \( v_ = 60\, \text{ft/s} \) - Car **B** decelerates at \( a_B = 12\, \text{ft/s}^2 \) - Car **A** decelerates at \( a_A = 18\, \text{ft/s}^2 \) (after reaction) - Car **A**'s reaction time, \( t_r = .75\, \text{s} \) - Distance between cars: \( d \) (to be found or used in the answer) --- ## **Step 1: Find stopping distance for each car** ### **Car B (Front Car)** Uses kinematic equation: \[ v^2 = v_^2 + 2a x \] Since \(v = \) (car stops), solve for \(x_B\): \[ = (60)^2 + 2(-12)x_B \] \[ = 360 - 24x_B \] \[ 24x_B = 360 \] \[ x_B = \frac{360}{24} = 150\, \text{ft} \] --- ### **Car A (Rear Car)** #### **Step 1a: Distance traveled during reaction time** During reaction time, car A continues at 60 ft/s for .75 s: \[ x_{A,\text{reaction}} = v_ t_r = 60 \times .75 = 45\, \text{ft} \] #### **Step 1b: Braking distance after reaction** Now, car A brakes with \( a_A = 18\, \text{ft/s}^2 \): \[ = (60)^2 + 2(-18)x_{A,\text{brake}} \] \[ = 360 - 36x_{A,\text{brake}} \] \[ 36x_{A,\text{brake}} = 360 \] \[ x_{A,\text{brake}} = \frac{360}{36} = 100\, \text{ft} \] #### **Total distance traveled by Car A before stopping:** \[ x_A = x_{A,\text{reaction}} + x_{A,\text{brake}} = 45 + 100 = 145\, \text{ft} \] --- ## **Step 2: Minimum safe distance \( d \)** For car A to avoid collision, it must not travel farther than car B plus the initial gap: \[ d + x_B \geq x_A \] \[ d \geq x_A - x_B \] \[ d \geq 145 - 150 = -5\, \text{ft} \] A negative value means that **even if car A was 5 ft behind car B, it would *just* avoid collision**. But for safety, the minimum distance should be ** ft**, i.e., cars are just about to touch. --- ## **Final Answer** \[ \boxed{d = \, \text{ft}} \] **This is the minimum safe distance required between the cars to avoid a collision, given the stated conditions.** If you want a safety margin, add a positive value to this result. --- ### **Summary Table** | Car | Total Stopping Distance (ft) | |-------------|-----------------------------| | Car B | 150 | | Car A | 145 | --- **Let me know if you need a more general formula or have specific values to use!**