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Unformatted text preview: Reference Answer of Assignment 1 ( SCI 3510 Mathematical and Statistical Modelling ) (1) Prove the equation of state for variable velocity u ( x, t ) and density ρ ( x, t ). (2) Suppose that the velocity u ( x, t ) is selected as the prime variable. Find the relation between flux q ( x, t ) and velocity u ( x, t ) for Greenshields model. Then prove the identity: U max parenleftbigg ρ ρ 2 ρ max parenrightbigg = ρ max parenleftbigg u u 2 U max parenrightbigg , Solution From the Greenshields model, we have u ( ρ ) = U max parenleftbigg 1 ρ ρ max parenrightbigg The above relation may be written as ρ ( u ) = ρ max parenleftbigg 1 u U max parenrightbigg Thus we have q ( x, t ) = u ( ρ ) · ρ = U max parenleftbigg ρ ρ 2 ρ max parenrightbigg or q ( x, t ) = u · ρ ( u ) = ρ max parenleftbigg u u 2 U max parenrightbigg The formula given in this problem is clearly an identity. (3) It is officially suggested by the Transportation Department of US to keep three carlength apart from the front car for a safety drive. Show that the maximum flux reachesfrom the front car for a safety drive....
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 Spring '11
 wong

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