1) A car is lapping an elliptical racetrack at a constant speed of 250. km/h in

Question

1) A car is lapping an elliptical racetrack at a constant speed of 250. km/h in the counterclockwise direction. The length of the major axis of the track is 1.60 km and the length of the minor axis is 0.800 km. The circumference of the racetrack is 3.87km. a) What is the average velocity of the car between points A and B? b) What is the average acceleration of the car between points A and B? c) At point A, the effective radius of the track is 1.60 km, and at point B, the effective radius of the track is 0.200 km. What is the minimum static coefficient of friction between the car's tires and the track if the car circles the track without skidding? (What I'm calling the effective radius is mathematically what's known as the radius of cur- vature. It means that because of how sharply the track is curved at point A, it's as if it were traveling around a circle of radius 1.60 km; and because of how sharply the track is curved at point B, it's as if it were traveling around a circle of radius 0.200 km.) 0.80 km 1.60 km Figure 1 - An elliptical racetrack 1.60 km long and 0.800 km wide.

Explanation / Answer

1. speed, v = 250 km/h = 69.4444 m/s

major axis, 2a = 1.6 km

minort axis, 2b = 0.8 km

circunference, C = 3.87 km

a. between A and B

average velcoity = displacement / time

displacement = sqroot(b^2 + a^2) = 0.8944 km = 894.42719 m

time = C/4v = 3870/4*69.444 = 13.932 s

average velocity = 64.199 m/s = 231.1164 km/hr

b. average acceleration = a

initial velocity = -69.444 j m/s

final velocity = 69.444 i m/s

time t = 13.932 s

69.444 i = -69.444 j + a*13.932

a = 69.444( i + j)/13.932

|a| = 7.0491 m/s/s

c. coefficient of static friction = k

then for a turn of radius r

v^2/r = k*g

r min = 200 m

69.44^2/200 = k*9.81

k = 2.457

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