Doing The Calculations

Take a sheet of paper, or if you prefer, a computerized spreadsheet.

Make eleven columns.

Column 1: Velocity

Fill in the column with values ranging from 0 to 120 feet-per-second in steps of 5. Thus, 0, 5, 10, 15, 20, and so on.

Column 2: Car length

Fill in this column with the value 12 (feet) as the length of the vehicles we are considering.

Column 3: Driver Reaction Time

Fill in this column with 1.2 (seconds).

Column 4: Braking acceleration

Fill in this column with 30 (fps/second)

Column 5: Time

Our first calculation!

Fill in each cell of the column with the value of the velocity divided by the braking acceleration, that is, column 1 divided by column 4.

Column 6: Braking distance

Fill in each cell of this column with the results of calculating half the braking acceleration multiplied by the square of the time.

That is, half multiplied by column 4 multiplied by the square of column 5.

Column 7: Space required for vehicle

Cells in this column are the sum of (a) the car length (b) the braking distance and (c) the driver reaction time multiplied by the speed.

That is, column 2 plus column 6 plus the product of columns 3 and 1.

Column 8:

Here we approach our solution to the problem of getting the most people home in the shortest time.

Column 8 will hold the number of vehicles per second that pass any given point on the highway.

We know the space required by a vehicle; another word for space is distance.

We know the speed of the vehicles.

We divide the speed (column 1) by the space (column 7) to find out how many vehicles per second pass a point on the highway.

For example, if a vehicle needs 74 feet of space and is traveling at 35 fps, then we can fit 35/74 parts of a vehicle on the highway every second.

Column 9:

This column serves to identify the maximum value of column 8, that is, the maximum number of vehicles for a range of speeds.

In the diagram reproduced below you will see that at a speed of 20 fps, we can fit 0.4348 vehicles per second, and that this is the maximum value for number of vehicles for our range of speeds.

Columns 10 and 11:

These columns are loaded with the value of column 1 (speed in fps) expressed in mph and Km/h for those of us accustomed to those units.

We get the most people home in the shortest time when we travel at 14 mph or 22 Km/h.

Of course, that is for 12-foot vehicles with braking acceleration of 20 fps/sec, and so on.

My Table Of Calculations

A Chart

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