To determine how far apart the cars end up if they manage to miss each other, we can approach the problem by calculating the stopping distance for each car and then comparing it to the initial distance between the cars.

Given:

• Initial velocity of both cars ($v_0$): $30 \, \text{m/s}$
• Deceleration ($a$): $8.5 \, \text{m/s}^2$
• Initial distance apart: $65 \, \text{m}$
• Mass of each car: $1200 \, \text{kg}$ (not needed for the calculation as it doesn't affect the stopping distance)

Step 1: Calculate the stopping distance for each car

The stopping distance ($d$) can be calculated using the kinematic equation: $v^2 = v_0^2 + 2ad$ Since the final velocity ($v$) is $0 \, \text{m/s}$ when the car stops: $0 = v_0^2 + 2ad$ Solving for $d$: $d = -\frac{v_0^2}{2a}$ Substituting the values: $d = -\frac{(30)^2}{2(-8.5)} = \frac{900}{17} \approx 52.94 \, \text{m}$

Step 2: Determine the total distance covered by both cars before they stop

Since both cars decelerate equally and start at the same speed, each car will travel the same distance before stopping. Therefore, the total distance both cars cover together is: $\text{Total distance covered} = 2 \times 52.94 \, \text{m} \approx 105.88 \, \text{m}$

Step 3: Compare the total stopping distance to the initial distance apart

The initial distance between the cars is $65 \, \text{m}$, which is less than the total distance they would have traveled to stop ($105.88 \, \text{m}$).

This means that if they were going to stop without collision, they would need more than 65 meters. However, since we are asked to find out how far apart they are if they manage to miss each other, the approach changes to finding out how far they are from each other once they reach their minimum distance.

Step 4: Calculate the remaining distance between the cars

Since the total stopping distance exceeds the initial separation, the cars will meet, so the calculation stops here. However, the question is hypothetical. If they were supposed to "just miss each other," that would mean the exact point at which they stop, or the closest they could theoretically get before stopping completely, would be 0 meters apart.

But if somehow they managed to avoid a collision by stopping exactly at the closest point without touching, they would be at:

0 meters apart.

Final Answer:

If the cars manage to avoid a collision in this hypothetical scenario, they would end up 0 meters apart, meaning they would stop right next to each other.

Would you like any details or have further questions? Here are some related questions you might find interesting:

1. What if the initial speeds were different for the two cars?
2. How would the problem change if the cars were on a slope?
3. How can you calculate the impact force if the cars did collide?
4. How does mass affect the stopping distance in real-world scenarios?
5. How would reaction time before braking affect the outcome?

Tip: In collision problems, it's essential to compare the stopping distances and relative initial distances to determine if the objects will collide or avoid each other.