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# Differential Equation Question watch

1. Its my 18th birthday today, and instead of celebrating i have to sit here revising. So to make my life easier, can someone figure this out:

This is a seperation of variables question - so dont try any other methods

A particle of mass m in free fall experiences air resistance proportional to it's velocity, v. It reached terminal velocity at V.

(i) Find the constant of proportionality.
Easy: terminal velocity so kV=mg, so k=mg/V

(ii) The particles is now projected upwards and initial speed U. Find an expression for acceleration (dv/dt):
Simple again: If upwards is positive v, then force on particle = -(mg + mgv/V) since the two forces act downwards. Divide through by m to find dv/dt: dv/dt = -(1 + v/V)g

(iii) Using dv/dt = vdv/dx, where x is height above the ground, find an expression for the maximum height of the particle

H = (V^2/g)( (U/V) - ln(1+U/V) )

any clues?
2. v/(V + v) dv/dx = -g/V
int v/(V + v) dv between U and 0 = int -g/V dx between 0 and H
int (1 - V/(V + v)) dv between U and 0 = -gH/V
[v - V*ln|V + v|] between U and 0 = -gH/V
-gH/V = U - V*ln(V + U) +V*lnV = V[U/V + ln[V/(V+U)]]
H = -V^2/g*[U/V - ln(1 + U/V)]
So it's just the negative sign I have a problem with, not sure where I should lose it though.
3. how comes you can turn v/(1+v/V) into v/(v+V)!"!??
4. ignore me...i see you left the V on the other side
5. Is this an M4 question? I'm not bothering revising for it yet as its still ages away
6. yea it's m4 - but it's only next wednesday (MEI m4)
7. (Original post by Willa)
yea it's m4 - but it's only next wednesday (MEI m4)
Ah right, edexcel is 1st july

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Updated: June 18, 2004
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