Question

Near the end of a marathon race, the first two runners are separated by a distance of 45 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?

Short answer

(a)0.7m/s

(b) The second runner will win the race.

(C) The winner will be 4 meter ahead.

Step-by-step solution

Definition of velocity

The directional speed of an item in motion is an indication of its rate of change in position as seen from a specific point of view and measured by a specific time standard.

Given data

The velocity of the second runner relative to earth is given$V_{se}=4.2m/s$

The velocity of the first runner relative to earth is $V_{ef}=3.50m/s$

The gravitational acceleration is -9.8m/s²

Velocity of the second runner relative to that of first runner

a. The velocity of the runner relative will be

$\begin{array}{ll}{v}_{sf}=V_{se}++v_{ef}& \\ v_{sf}=(4.2)+(-3.5)& |V_{ef}=-V_{fe}]\\ v_{sf}=0.7m/s& \end{array}$

The velocity of the second runner relative to first runner 0.7 m/s.

Who will win the race

b. The distance travelled by the second runner

$$\begin{gathered} V=4.2m/s \\ d=45+250 \\ d=295m \end{gathered}$$

Using the formula to find the time

$\begin{array}{l}v=\frac{x}{t}\\ 4.2=\frac{295}{t}\\ t=70.2s\end{array}$

The time taken by the second runner is 70.0 s.

The distance travelled by the first runner

V = 3.5 m/s

d = 250 m

Using the formula to find the time

$\begin{array}{l}v=\frac{x}{t}\\ 35=\frac{250}{t}\\ t=71.4s\end{array}$

The time taken by the second runner is 71.4 s

Hence the second runner will win the race who has speed 4.2 m/s.

Step 3: Distance ahead will the winner be when she crosses the finish line?

c. Distance ahead will the winner be when she crosses the finish line?

$\begin{array}{l}\overline{V}=\frac{x}{t}\\ 3.5=\frac{x}{70.2}\\ x=246m\end{array}$

Hence the winner will be 250-246 = 4 meter ahead.