Physics problem

Question

A cross-country rally car driver sets out on a 100.0 km race. At the halfway marker (50.0 km), her pit crew radios that she has averaged only 80.0 km/h. How fast must she drive over the remaining of the distance in order to average 100.0 km/h for the entire race?

Answer 1

The definition of average velocity is (total displacement)/(total time).  All you look at is the displacement between where you start and where you finish (in this case, a magnitude of 100 km), and divide it by the time it takes to complete that displacement.

 

To average 100 km/hr in a 100 km race, you must (as you can probably figure out) complete the race in 1 hour.

 

What is the time the racer has taken to go the first 50 km?  We can calculate that with dimensional analysis.  If the racer is going 80 km/hr, that also means it takes her 1 hr to go 80 km.  To figure out how long it takes to go 50 km at that speed, use:

 

50 km * (1 hr/80 km) = 5/8 hour

 

We know the total time to complete the race and average 100 km/hr is 1 hr.  Thus, she has (1 hr - 5/8 hr) = 3/8 hr to complete the second 50 km to meet this goal.

 

How fast does someone have to go to cover 50 km in 3/8 hr?  v= 50km/(3/8 hr) = 50*(8/3) = 133.3 km/h

 

Having this average velocity over the second 50 km will allow the racer to average 100 km/h for the entire race.

Answer 2

Let t1 = time to reach x1 = 50 km while travelling at speed v1 = 80 km/h

 

t1 = x1/v1 = 50/80 h = 0.625 h

 

Let t2 = time to travel distance x2 = 50 km (from the end of the first 50 km to the end of the trip at 100 km)

 

t2 = x2/v2 = 50/v2

 

But t1 + t2 = 1 h (total trip of 100 km must be completed in 1 hr)

 

t2 = 1 - t1 = 1 - 0.625 = 0.375 h

 

t2 = x2/v2 ⇒ v2 = x2/t2 = 50/0.375 = 133.33 km/h