To set a speed record in a measured (straight-line) distance d, a race car must

Question

To set a speed record in a measured (straight-line) distance d, a race car must be driven first in one 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, what number of method should we use: the average of d/t1 and d/t2 (method 1) or should we divide d by the average of t1 and t2 (method 2)? (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?

Explanation / Answer

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

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