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 proportional 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, let's break down the information step by step:
1. Driver A had been leading archrival B by a steady 3 miles.
2. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at a rate that is proportional to the square of his remaining speed.
3. One mile after running out of gas, driver A's speed was exactly halved.
4. Driver B's speed remained constant.
Let's solve this step by step:
Step 1: Driver A had been leading by a steady 3 miles.
Since driver A was leading archrival B by 3 miles, this means that the distance between them was 3 miles at that point.
Step 2: Driver A ran out of gas and decelerated thereafter.
After running out of gas, driver A started to decelerate at a rate proportional to the square of his remaining speed. This means that the rate of deceleration is related to the square of his speed.
Step 3: One mile after running out of gas, driver A's speed was exactly halved.
After one mile of running out of gas, driver A's speed was halved. Let's say his speed at that point was V, then his speed after one mile would be V/2.
Step 4: Driver B's speed remained constant.
Since no information is given about driver B's speed changing, we can assume that driver B's speed remained constant throughout the race.
Considering all the steps, it is evident that driver A's speed continued to decrease after running out of gas, while driver B's speed remained constant. Thus, driver B would have taken the lead and won the race.
To determine who won the race, we need to find out if driver A or driver B crossed the finish line first. Let's break down the information given and analyze the scenario step by step.
1. Initially, driver A had been leading driver B by a steady 3 miles.
2. Two miles from the finish line, driver A ran out of gas and started decelerating. The deceleration is proportional to the square of driver A's remaining speed.
3. One mile later (1 mile from the finish line), driver A's speed was halved.
4. Driver B's speed remained constant.
Now, let's calculate the remaining speed of driver A at each of the given points to see if we can determine the winner.
- Two miles from the finish line:
Since driver A ran out of gas at this point, we can assume that his speed reduced continuously until he halted completely. However, since we are only concerned about the relative speeds of the drivers, we can skip the deceleration calculation and directly evaluate the remaining speeds.
- One mile from the finish line:
At this point, driver A's speed is halved compared to his initial speed.
To simplify the calculation, let's assume driver A's initial speed was "S" miles per hour (mph). Therefore, at one mile from the finish line, driver A's speed is "S/2" mph.
Now, let's consider driver B's speed. Since it's mentioned that driver B's speed remained constant throughout the race, we can assume it also remains constant near the finish line.
So, at one mile from the finish line, driver B's speed is constant and remains the same as during the entire race.
Now, let's analyze the final stretch of the race, which is 1 mile long.
If driver A's remaining speed is halved, then driver B would have to cover twice the distance while maintaining his constant speed to beat driver A.
Therefore, the conclusion is that driver B won the race.
To summarize:
Driver B won the race because driver A, after running out of gas, decelerated and had his speed halved one mile before the finish line, giving driver B an opportunity to overtake him.