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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 car-length 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