Model for single lane of traffic

[andrey21]

A model for a single lane of traffic is given by the following pde

p dv/dx + v dp/dx + dp/dt = 0

Where:

v = kx/p



Show that

dp/dt = -k



Here is my attempt

v = kx/p

dv/dx = k/p

p= kx/v

dp/dx = k/v

Substituting into original pde:

p (k/p) + v(k/v) + dp/dt=0

Clearly made an error, any help would be great

 

[elphin]

p dv/dx + v dp/dx + dp/dt = 0

Where:

v = kx/p

Show that

dp/dt = -k

p dv/dx + v dp/dx + dp/dt = 0 can be written as

d(pv)/dx + dp/dt = 0 ----------- 1

v = kx/p => pv = kx -> put in 1

we get dp/dt = - k

method you used is wrong because you can't partially differentiate v w.r.t x in an equation which contains p (also a function of x) -. while differentiating you treated p as a constant w.r.t to x..