Near the end of a marathon race, the first two runners are separated by a distance of 45.0 m. The front runner has a velocity of 3.50 m/s, and the second a velocity of 4.20 m/s. (a) What is the velocity of the second runner relative to the first? (b) If the front runner is 250 m from the finish line, who will win the race, assuming they run at constant velocity? (c) What distance ahead will the winner be when she crosses the finish line?

Answer:a) 0.75m/s

b) The second runner will win.

c) The second runner will go ahead of the first runner in the next 43m.

Explanation:

Velocity of first runner, V1 = 3.45m/s

Velocity of second runner, V2 = 4.20m/s

Distance between the two runners = 45m

a) V1,2 = V2-V1 = (4.20-3.50) m/s

V1,2 = 0.75m/s

Time taken to cover 45m

t = d/v = 45/0.75 = 60seconds

Distance covered by first runner, d1 = V1 * t

d1 = 3.45m/s * 60sec = 207m

Distance covered by second runner, d2 = 4.20m/s * 60secs = 252m

The second runner will win.

c) Length of race=250m

The second runner's distance is 252m

The second runner will go ahead of the first by (250-207) m=43m

a. 0.7 m/s

b. The second runner

c. 4.20 meters

Explanation:

a. To know the relative speed we can use the following equation:

Relative Speed (Vr) = V1 - V2

The relative speed is used to compared the velocity of one object (V1) to another object (V2). To learn the Relative Velocity from the second runner to the first we do the following:

Relative Speed (Vr) = Second Runner Velocity (Vs) - First Runner Velocity (Vf)

Vr = 4.20 m/s - 3.50 m/s

Vr = 0.7 m/s

This means that the second runner is gaining 0.7 meter from the first runner each second.

b. To know which runner will gain the race, we need to keep in mind, that the seconds runner is 45 meter more away that the first runner from the goal. Knowing this, the winner will be the one that reach the goal on less time.

Because movement is constant, we can use the following equation:

Velocity (V) = Distance (D) / Time (T)

To get the time we do:

T = D / V

Now, for the first runner:

T = 250 meters / (3.50 m/s)

T = 71.42 seconds

For the second runner.

T = (250 meters + 45 meters) / (4.20 m/s)

T = 70,23 seconds

Because the second runner takes less time to reach the goal, he wins the race.

c. Because it is know how much time takes the winner to get to the finish line, we can use that same time to know how far was the first runner from the goal. This is posible we know that how far he is from the goal is as follows.

Distance from goal (Dg) = Total Distance (D) - Distance travelled (Dt)

Where Total Distance (D) is 250 mts. And the distance travelled can be obtained by using the following equation.

Velocity (V) = Distance (D) / Time (T)

Searching for distance:

D = V / T.

D = (3.50 m/s) * 70.23 s

D = 245,80 meters

Soo

Distance from goal (Dg) = Total Distance (D) - Distance travelled (Dt)

Dg = 250 meters - 245,80 meters

Dg = 4,20 meters

The first runner was 4,20 meter behind the second one when the second runner crossed the goal