Question

On a trip, you get stuck in slow traffic and have to drive 90.0 km at 30.0 km/hr. Then the traffic clears up, and you're able to drive another 90.0 km at 60.0 km/hr.

a) Calculate your overall average speed for this 180.0 km trip. Check: it's not 45.0 km/hr! See the next question as well. Use the definition of average speed in physics.

b) Prove that if you'd driven 180. km at 60.0 km/hr for the second leg, your overall average speed would have been 45.0 km/hr.

c) In the original description of the trip, why was your overall average speed lower than 45.0 km/hr, the mathematical average of 30.0 km/hr and 60.0 km/hr?

d) Determine the speed at which you'd have to have driven the second 90.0 km so that your overall average speed would have been 45.0 km/hr.

e) What are the largest and smallest possible average velocities for this trip, as originally described? What further information would allow you to determine where in this range your actual average velocity was? Explain in practical terms.

Answer 1

ans. (a) speed of car for 1st 90 km = 30 km/h

speed of car for 2nd 90 km = 60 km/h

time taken = distance / speed

time taken to cover 1st 90 km, t = (90 / 30 )  

= 3 hr

time taken to cover 2nd 90 km, T = 90 / 60

= 1.5 hr

Average speed = Total distance / Total time

= 180 / (3+1.5)

= 40 km/hr

therefore, the average speed in covering 180 km is 40 km/hr.

(b) average speed for the trip of 180 km = 45 km/hr

average speed for 2nd 90 km = 60 km/hr

  time taken to cover 2nd 90 km = 90 / 60

= 1.5 hr

   time taken to cover 1st 90 km = t

  Average speed = Total distance / Total time

45 = Total distance / (t + T)

45 = 180 / ( t + 1.5)

45 ( t+ 1.5) = 180

45 t + 45 (1.5) = 180

45 t + 67.5 = 180

45 t = 180 - 67.5

t = 112.5 / 45

t = 2.5 hr

average speed for 1st 90 km = 90 / 2.5

= 36 km/hr

Therefore, for average speed to be 45 km / hr for 180 km at 60 km / hr for 2nd 90 km, the average speed for the 1st 90 km should be 36 km/ hr.

(c) The average speed is the total distance covered divided by the total time taken and we have calculated in option (a) that average speed for the trip is 40 km/hr. It cannot be 45 km/hr ( average of 30 km/hr and 60 km/hr ) because the time taken in 1st 90 km is different from the time taken during covering 2nd 90 km.

(d) average speed for the trip of 180 km = 45 km/hr

average speed for 1st 90 km = 30 km/hr

  time taken to cover 1st 90 km, t = 90 / 30

= 3 hr

   time taken to cover 2nd 90 km = T

  Average speed = Total distance / Total time

45 = Total distance / (t + T)

45 = 180 / ( 3 + T)

45 ( 3+ T) = 180

45 T + 45 (3) = 180

45 t + 135 = 180

45 t = 180 - 135

t = 45 / 45

t = 1 hr

average speed for 1st 90 km = 90 / 1

= 90 km/hr

  Therefore, for average speed to be 45 km / hr for 180 km at 30 km / hr for 1st 90 km, the average speed for the 2nd 90 km should be 90 km/ hr.