To set a speed record in a measured (straight-line) distance d, a race car must be driven first in one ?

Question

direction (in time t1) and then in the opposite direction (in time t2). (a) to eliminate the effects of the wind and obtain the car's speed vc in a windless situation, should we find the average of d/t1 and d/t2 (method 1) or should we divide d by the average of t1 and t2? (b) What is the fractional difference in the two methods when a steady wind blows along the car's route and the ratio of the wind speed vw to the car's speed vc is 0.0240?

Answer 1

v = speed of car with no wind

u = speed of the wind along the path of the car

v - u = speed when going against the wind

v + u = when going in the same direction as the wind

v - u = d/t1

v + u = d/t2

2v = [d/t1 + d/t2]

v = (1/2)[d/t1 + d/t2]

Of course the meanings of t1 and t2 are not important since they appear in the equation in exactly the same way.So the method to use is method 1.

u/v = 0.0240 so u = 0.024v

Other method v = d/[(t1 + t2)/2] = (2){1/[t1/d + t2/d]}

diff = (1/2)[d/t1 + d/t2] - (2){1/[t1/d + t2/d]}

diff = (1/2){d/t1 + d/t2 - 4/[t1/d + t2/d]}

diff = (1/2){2v - 4/[1/(v - u) + 1/v + u)]}

diff = (1/2){2v - 4(v + u)(v - u)/2v}

diff = [1/(4v)][4v^2 - 4(v^2  - u^2)]

diff = (u)(u/v) = u^2/v

fractional diff = diff/v = (u/v)^2 = (0.024)^2 = 0.000576