Driver A had been leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at a propertional to the square of his remaining speed. One mile later, driver A's speed was exactly halved.If driver B's speed remained constant, who won the race?
To determine who won the race, we need to analyze the distance covered by each driver after driver A ran out of gas.
Let's break down the given information:
1. Initially, driver A was leading driver B by a steady 3 miles.
2. Driver A decelerated after running out of gas, with the deceleration being proportional to the square of their remaining speed.
3. One mile after running out of gas, driver A's speed was exactly halved.
4. Driver B's speed remained constant.
To find the solution, we need to determine how far driver A and driver B traveled after driver A ran out of gas.
Let's start by finding out how far driver A traveled after running out of gas:
1. Since driver A was leading driver B by 3 miles, it means they covered an additional 3 miles before running out of gas.
2. After running out of gas, driver A continues to decelerate at a rate proportional to the square of their remaining speed.
3. One mile after running out of gas, driver A's speed is halved. This means that the remaining distance driver A covers will also be halved compared to their initial speed.
4. If driver A's speed is halved after one mile, it means they would have traveled one mile at their initial speed.
5. Let's assume driver A's initial speed when they ran out of gas was V mph. After one mile, driver A's speed reduced to V/2 mph.
6. To calculate the distance traveled by driver A after running out of gas, we need to determine the time taken. As speed is distance divided by time, we can rearrange the formula to time = distance / speed.
7. The time taken by driver A to cover the remaining distance after one mile can be calculated as (2 miles - 1 mile) / (V/2 mph) = 1 / (V/2) hours = 2 / V hours.
8. Now, we can calculate the distance traveled by driver A after running out of gas using the formula distance = speed * time. The distance traveled by driver A after running out of gas is (V/2 mph) * (2/V hours) = 1 mile.
Therefore, after running out of gas, driver A covers an additional 1 mile before their speed becomes zero.
Next, let's consider driver B:
1. Driver B's speed remained constant throughout the race.
2. Since driver A ran out of gas only 2 miles from the finish, the remaining distance for driver B to cover is 2 miles.
Hence, driver B covers the remaining 2 miles without any change in speed.
To determine who won the race, we need to compare the total distance covered by each driver:
- Driver A: Initial distance (3 miles) + Extra distance covered after running out of gas (1 mile) = 3 + 1 = 4 miles.
- Driver B: Remaining distance to the finish line (2 miles) = 2 miles.
Therefore, driver A covered a total distance of 4 miles, while driver B covered only 2 miles. Driver A won the race.