John and Al are in a 15 km race. John averages 4.4 m/s during the first half of the race and then runs at a speed of 4.2 m/s until the last 200 m, which he covers at 4.5 m/s. At what average speed must Al run to beat John?
To determine the average speed Al must run to beat John, we first need to find the total time it takes for John to complete the race. Then, we can calculate the required average speed for Al by dividing the total distance by the total time.
First, let's break down John's race performance into three parts:
1. John's speed during the first half of the race (7.5 km):
Distance = 7.5 km
Speed = 4.4 m/s
Time = Distance / Speed = 7,500 m / 4.4 m/s
2. John's speed from the halfway point to the last 200 m (7.5 km):
Distance = 7.5 km
Speed = 4.2 m/s
Time = Distance / Speed = 7,500 m / 4.2 m/s
3. John's speed for the last 200 m:
Distance = 200 m
Speed = 4.5 m/s
Time = Distance / Speed = 200 m / 4.5 m/s
Now, we can add up the three times to find the total time it takes for John to complete the race:
Total Time = Time (First half) + Time (Second half) + Time (Last 200 m)
Next, we need to calculate the average speed Al must run to beat John. The total distance of the race is 15 km.
Average Speed for Al = Total Distance / Total Time
Now, let's substitute the values and calculate:
Total Time = (7.5 km / 4.4 m/s) + (7.5 km / 4.2 m/s) + (200 m / 4.5 m/s)
Average Speed for Al = 15 km / Total Time
By plugging in the calculated value for the total time, you will find the average speed Al must run to beat John.