Final answer:
The relative velocity of the second runner with respect to the first is 0.70 m/s. The second runner will overtake the front runner before the finish line and win the race, finishing 25.3 meters ahead.
Step-by-step explanation:
Calculating Relative Velocity and Race Outcome
To find the relative velocity of the second runner with respect to the first (a), we subtract the velocity of the front runner from the velocity of the second runner. Therefore, the relative velocity is:
Relative velocity = 4.20 m/s - 3.50 m/s = 0.70 m/s.
Calculating who wins the race (b), we set up equations for the distance each runner will cover in a given time t, assuming constant velocity:
- Front runner: Distance = velocity * time = 3.50 m/s * t
- Second runner: Distance = 45.0 m (gap) + (velocity * time) = 45.0 m + 4.20 m/s * t
The second runner catches up when their distances are equal:
45.0 m + 4.20 m/s * t = 3.50 m/s * t
Solving for t gives us t = (45.0 m) / (0.70 m/s) = 64.3 seconds.
Since the front runner is 250 m from the finish, the time it takes for the front runner to finish is:
Time = distance / velocity = 250 m / 3.50 m/s = 71.4 seconds.
The second runner will therefore win the race, as they will catch the front runner at 64.3 seconds and both have enough distance to reach the finish line.
Finally, calculating the winning distance ahead (c), when the second runner crosses the finish line, the first runner will have:
Distance covered by first runner in 64.3 seconds = 3.50 m/s * 64.3 s = 224.7 m.
Therefore, the winner will be:
Winning distance ahead = 250 m - 224.7 m = 25.3 meters.