How to Push a Car?

A man pushing his car

There are only two classes of pedestrian in these days of reckless motor traffic – the quick and the dead.

           Lord Dewar

There was an old joke that asked, ‘Why does a Lada car have a heater on its rear windscreen?’ The answer: ‘So that your hands don’t get cold while you are pushing it.’ But pushing cars presents an interesting problem. Suppose you have to push your car into the garage and bring it to a stop before it hits the back wall. How should you push it and pull it so as to get into the garage and stopped as quickly as possible?  

        The answer is that you should accelerate the car by pushing as hard as you can for half the distance to be covered and then decelerate it by pulling as hard as you can for the other half. The car will begin stationary and finish stationary and take the least possible time to do so.

        This type of problem is an example of an area of mathematics called ‘control theory’. Typically you might want to regulate or guide some type of movement by applying a force. The solution for the car-parking problem is an example of what is called ‘bang-bang’ control. You have just two responses: push, then pull. Domestic temperature thermostats often work like that. When the temperature gets too high they turn on cooling; when the temperature gets too low they turn on heating. Over a long period you get temperature changes that zigzag up and down between the two boundaries you have set. This is not always the best way to control a situation. Suppose you want to control your car on the road by using the steering wheel. A robot driver programmed with the bang-bang control approach would let the car run into the left-hand lane line, then correct to head to the right until it crossed the right-hand lane line, and so on, back and forth. You would soon end up being stopped and invited to blow into a plastic tube before being detained in the local police cells if you followed this bang-bang driving strategy. A better approach is to apply corrections that are proportional to the degree of deviation from the medium position. A swing seat is like this. If it is pushed just a little way from the vertical then it will swing back more slowly than if it is given a big push away from the vertical.

A man pushing his car

        Another interesting application of control theory has been to the study of middle- and long distance running – and presumably it would work in the same way for horse racing as well as human racing. Given that there is only a certain amount of oxygen available to the runner’s muscles, and a limit to how much it can be replenished by breathing, what is the best way to run so as to minimize the time taken to complete a given distance? A control theory solution of bang-bang type specifies that for races longer than about 300 metres (which we know is where anaerobic exercise is beginning and oxygen debt begins) you should first apply maximum acceleration for a short period, then run at constant speed before decelerating at the end for the same short period that you accelerated for initially. Of course, while this may tell you how best to run a time-trial, it is not necessarily the best way to win a race where runners are competing against you. If you have a fast finish or have trained to cope with very severe changes of pace you may well have an advantage over others by adopting different tactics. It will be a brave competitor who sticks to the optimal solution while others are racing into a long lead. Sitting in behind the optimal strategist, sheltering from the wind and getting a free ride before sprinting to victory in the final straight is a very good alternative plan.

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