Question #4320

hi i have a question for you,
To set a speed record in measured straight line distance d, a race car must be driven first in one direction in time t1 and then in the opposite direction time t2. a) to eliminate the effects of the wind and obtain the ca's speed Vc in a windless situation, should we find the average of d/t1 and d/t2 (method 1) or should we devidee d by the average of t1 and t2. b) what is the fractional difference in the two methods when a steady wind belows along the car's route and the ratio of the wind speed Vw to the car's speed Vc is 0.0180

Expert's 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 = 0.018 so u = 0.018v

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 = (0.018)^2 = 0,00032

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 = 0.018 so u = 0.018v

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 = (0.018)^2 = 0,00032

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