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F1 Racing Module

Story:

In the midst of a high-stakes Formula 1 Grand Prix race, two rival teams anxiously monitored the heated battle unfolding between their top drivers, Car A and Car B, as they hurtled down the 5 km straight circuit.

Although Car A had secured an early lead, the telemetry data revealed that Car B possessed a superior acceleration capability, slowly closing the gap with each passing lap. As the race reached its climactic moments, the teams' strategists faced a critical decision – to determine the precise moment when Car B's relentless pursuit would culminate in an electrifying overtake, seizing the lead and potentially securing victory.

Problem Statement:
Two Formula 1 cars, Car A and Car B, are racing on a straight 5 km (5000 meters) track. At the start of the race, Car A has a lead of 500 meters over Car B. However, Car B has a higher acceleration capability. The problem is to determine the exact time when Car B will overtake the leading Car A, and their respective positions at that moment.
Challenges:

At the start, Car A had a 500-meter lead over Car B. However, the equations governing their motion revealed Car B's higher acceleration capability. Car A's position was given by x_A = 1000 + 50t + 2t^2 meters, with an initial position of 1000 meters and starting speed of 50 m/s. Car B's position was x_B = 500 + 60t + 5t^2 meters, with a higher starting speed of 60 m/s, despite being 500 meters behind.

The crucial difference lay in their accelerations. Car A's acceleration was 4 m/s^2, while Car B boasted a remarkable 10 m/s^2 – more than double that of Car A. This meant Car B's velocity would increase at a much faster rate, allowing it to gradually close the gap.

The teams faced the challenge of determining the precise time when Car B's superior acceleration would enable it to overtake the leading Car A and their respective positions at that pivotal moment, which would ultimately decide the race outcome.

Or In simple terms, even though Car A starts 500 meters ahead, at what time will the faster accelerating Car B catch up and overtake Car A on the 5 km track? And what will be the position of both cars at that crucial moment?
Objective
Determine the exact time when Car B, with its higher acceleration, will overtake the initially leading Car A, and find their respective positions at that crucial moment in the race.
How to Provide a Solution:
1. Understand the Problem.
2. Carefully review the problem statement, story, and any additional information provided.
3. Identify the key constraints, challenges, and objectives.
4. Provide Your Solution with Calculations.
5. Propose a well-reasoned solution that addresses the challenges and constraints.
6. Support your solution with relevant calculations, considering factors related to problems.
7. Utilize the AI assistant to validate your assumptions, calculations, and approach.
Possible Solution Template.
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https://arcadebow.notion.site/SWAT-in-Action-Solution-54f43610e3ff48c083cbc6b155d69335

Solution