Q54PE - College Physics (Urone)

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?

Step-by-Step Solution

Answer

(a) 0.7m/s 

(b) The second runner will win the race. 

(C) The winner will be 4 meter ahead.

Step 1: 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.

Step 1: Given data

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

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

The gravitational acceleration is  -9.8m/s²

Step 2: Velocity of the second runner relative to that of first runner

a. The velocity of the runner relative will be

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

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

Step 3: Who will win the race

b. The distance travelled by the second runner

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

Using the formula to find the time

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

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?

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

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