Near the end of a marathon race, the first two runners are separated by a distance of 41.6 m. The front runner has a velocity of 3.4 m/s, and the second a velocity of 4.85 m/s. What is the magnitude of the velocity of the second runner relative to the first? If the front runner is 215 m from the finish line, who will win the race, assuming they run at constant velocity?
By what distance does the winning runner finish ahead of the next runner?

v₂₁ = 1.45 m/s

Second runner is the winner.

Δs = 35.1 m

Explanation:

For the relative velocity, we use the formula:

v₂₁ = v₂ - v₁

where,

v₂₁ = relative velocity of second runner with respect to first runner = ?

v₁ = velocity of first runner = 3.4 m/s

v₂ = velocity of second runner = 4.85 m/s

Therefore,

v₂₁ = 4.85 m/s - 3.4 m/s

v₂₁ = 1.45 m/s

Now, for finding the winner, we calculate the time taken by both the runners to reach finish line, by using following equation:

s = vt

t = s/v

for first runner:

t₁ = (215 m)/(3.4 m/s)

t₁ = 63.23 s

for 2nd runner:

t₂ = (215 m + 41.6 m)/(4.85 m/s)

t₂ = 52.9 s

Since, t₂<t₁.

Therefore, second runner is the winner.

Now, for the difference between runners at the time of winning, we first calculate the distance covered by first runner at that time. Using second equation of motion:

s = (3.4 m/s)(52.9 s)

s = 179.9 m

So, the distance by which the second runner finishes ahead of the first runner is given as follows:

Δs = 215 m - 179.9 m

Δs = 35.1 m