Driving habits, aerodynamic drag and horsepower

I touched on the subject of speed limits and speeding and drive time earlier, briefly. I wanted to revisit that. And you'll see why.

If we travel the same distance at two different speeds (velocities) we will record two different times.

Distance = velocity₁ × time₁ = velocity₂ × time₂

Let's say velocity₂ is twenty percent higher than velocity₁. Then you can write:

velocity₂ = 120% velocity₁ = 1.2 × velocity₁

So essentially you have a method to compare by how much time went down:

velocity₁ × time₁ = 1.2 × velocity₁ × time₂

Therefore: time₂ = 0.83 × time₁ = 83% time₁

You can save about 17% of time by going 20% faster.

I have done a few calculations, the up and down arrows signify by how much velocity goes up and by how much time goes down correspondingly:

Velocity ↑ 10%, time ↓ 9%

Velocity ↑ 15%, time ↓ 13%

Velocity ↑ 20%, time ↓ 17%

Velocity ↑ 25%, time ↓ 20%

Velocity ↑ 30%, time ↓ 23%

Velocity ↑ 35%, time ↓ 26%

Velocity ↑ 40%, time ↓ 29%

Velocity ↑ 45%, time ↓ 31%

Here's a more concrete example, showing the speed and the time taken to cover the same distance:

Going a distance of 25 miles @

65 mph : 23 min

70 mph : 21 min

75 mph : 20 min

80 mph : 19 min

85 mph : 18 min

Now that's one side of the equation: how much time you saved. What's the cost of that?

When you move through air, it pushes you back. That's called drag and the formula looks something like this:

Drag = ½ × Drag coefficient × Density of air × Area facing the wind × velocity ²

In that formula, the drag coefficient, the density of air and the area facing the wind all remain constant. The only variable there is velocity. The equation can be written as:

Drag = Constant factor × velocity ²

So the drag, which is a force experienced by a car, increases as the square of the velocity. Increase velocity by a factor of two, the drag will increase by a factor of four!

But wait, there's more. The power required to overcome this force and keep moving at that velocity is given by:

Power = Drag × velocity = Constant factor × velocity ² × velocity = Constant factor × velocity ³

What?! The power required now has velocity cubed as a factor. If you increase velocity by a factor of two, the power required to keep moving will increase by a factor of EIGHT!

Now, using the example above, an increase in speed from 65 mph to 85 mph, is roughly a 30% jump. Let's plug these numbers in and see how much power it takes to keep the car going:

Power₁ = Constant factor × velocity ³

Power₂ = Constant factor × (1.3 × velocity) ³

Power₂ = 2.2 × Power₁ = 220% Power₁

To drive at 85 mph, compared to 65 mph, the power required is more than double. Although engines have an optimal load point, for most engines that are in use, anything beyond 65 mph is inefficient. Also for most engines, horsepower and fuel economy are inversely related. In other words, horsepower and fuel consumption are directly proportional to each other.

By increasing your speed from 65 mph to 85 mph you gained 5 minutes. If this is your morning commute to work, isn't it easy just to start 5 minutes earlier? If you are in a job that is like everyone else's then those 5 minutes are probably not crucial. If you are in a job where it is crucial, like you are a captain on a submarine, then you probably don't drive to work, or don't have time to read this blog either, for that matter.