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