In Ireland, the offence of speeding is dealt with by means of a Fixed Charge system, which works as follows:
 The driver can, during the period of 28 days from the date of issue of the notice, make a payment of €80, or
 He/she can, during the next 28 days, make a payment of €120.
There are two flaws that I have identified with this penalty system
 The amount payable remains the same, regardless of how much over the speed limit the driver was going. In other words, the driver who was just over the limit gets the same treatment as the driver who was going well over it.
 The amount payable remains the same, regardless of the driver's income. The fine is €80, which can be a considerable amount of money for someone earning the bare minimum to get by, but it is peanuts for someone earning upwards of sixfigure sums.
First, let's consider kinetic energy. The kinetic energy of a moving body is proportional, not to its velocity, but to the square of its velocity. In other words:
where E = kinetic energy, in joules, m = mass, in kilograms, and v = velocity in metres per second.
Clearly, travelling at double the speed increases the kinetic energy not twofold but fourfold. It stands to reason that the fines should vary uniformly not with speed, but with the square of the speed, for example:
where f = fine payable, s = speed in kilometres per hour, and A, B and C are coefficients which would be determined not only by the speed limit but by other relevant factors, like proximity to a school, the condition of the road, or whether or not it was during a bank holiday weekend.
In order to determine a set of coefficients, for a certain situation, say, a speed limit of 60km/h, for cars, coming from a 100km/h zone into a 50km/h zone, we would need to set a fine for two speeds, say, €40 for 65km/h and €120 for 80km/h. We would also need a third parameter, which I call the slope. This is calculated by getting the derivative of the second equation above, as follows
where m = slope. We would typically set the slope at a certain value at the lower speed. With these parameters, we could put all these values into one equation, as follows:
where s1 and s2 are the lower and higher speeds respectively, f1 and f2 are the fines at s1 and s2 respectively, and m is the slope. To get the coefficients A, B and C, we would need to get the inverse of the 3x3 matrix, and rearrange the above equation as follows:
And that's it. Now to do an example. Let's say we could set a fine of €40 for 65km/h, €120 for 80km/h, and a slope of zero for the lower speed. Plug these values into the equation as follows:
Let's invert the 3x3 matrix and rearrange the equation:
We can now multiply the matrices on the lefthand side of the equals sign, to get the coefficients, and here are the values we get:
 A = 0.356
 B = 46.222
 C = 1542.222
Speed

Fine

65

€40.00

70

€48.89

75

€75.56

80

€120.00

85

€182.22

90

€262.22

95

€360.00

100

€475.56

105

€608.89

110

€760.00

115

€928.89

120

€1,115.56

125

€1,320.00

130

€1,542.22

135

€1,782.22

140

€2,040.00

These fines as calculated above would typically apply to someone with an annual gross income of €25,000. To get the final fine payable, we would simply multiply the initial fine by the driver's actual income, divided by 25000.
Let's say that a motorist who is earning €40,000 gross annually was caught doing 85km/h. From the table above, we can see that an 8km/h speed would result in a fine of €182.22. We would then make the incomebased adjustment by multiplying 40,000/25,000, and the final amount payable would be €291.55. Rounding it to the nearest whole euro, we get €292.00.