Car slowing due to air resistance

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joe
Posts: 749
Joined: Fri Aug 29, 2003 11:57 am
Location: Sydney

Car slowing due to air resistance

A student writes:

From the lecture notes: A trickier but real example. Over a marked kilometre, on a flat track with no wind, sUNSWift slows
from v0 = 70 kph to v = 50 kph. Assume that a = – kv2 (turbulent drag only). What is k?
This is what i did:

v0 =70kph, v = 50kph, x = 1km, x0 = 0km, a = -kv2, k = ?

2a (x – x0) = v2 – v02

2 (-k*(50)2)(1 – 0) = 502 - 702

k = 48e-2
The problem is that we derived
2a (x – x0) = v2 – v02
assuming constant a. In this problem, however, a = – kv^2, which is not constant. Because a is varying, we must integrate.

dv/dt = -kv^2
we want v and x, not v and t, so use dt = dx/v, which gives
v.dv/dx = - kv^2
now separate variables, i.e. put x on one side and v on the other
dv/v = - kdx
dx/x is an important combination to remember: it occurs a lot in physics. Substitute:
d(ln v) = - kdx
ln v - ln v0 = - kx
ln (v/v0) = - kx
v = v0 exp(-kx)

and finally remember that this is only the approximation for high speed. At low speed, rolling resistance dominates.
Joe