Car Parking summary March 2022.docx 08/03/22
ENG2012 MATHEMATICAL MODELLING – PARKING A CAR
Summary of case-study 7/8 March 2022
Aim
If a line of cars is parked by the side of a road, what is the shortest gap into which a car can be
backed?
Method
Calculate by driving out - starting position known. Single movement, wheels at full-lock.
R
2
R
W
D L
x
1
Lr
The diagram shows the starting position 1 of the car, and the position 2 after the first
movement, when the front corner of the car is just missing the rear corner of the car ahead. W
is the width of the car, and L is the length from the rear wheels to the front corner. Lr is the
distance from rear wheels to rear corner. D is the length of the gap ahead of the car. R is the
turning radius – interpreted here as the ‘wall-to-wall’ radius. x is the distance from the off-
side rear wheel to the centre of the turning circle (an unknown that must be eliminated).
We can use Pythagoras Theorem on the two right-angled triangles to get relationships
between W, D, L, x, R and hence find D in terms of W, L and R (eliminating x).
Assumptions and Simplifications
Car is a simple rectangle
Rear wheels roll forwards with no side-slip
Initial position is against the car behind and against kerb or wall
…… - many more simplifications and assumptions need to be identified, with an indication
which are unrealistic and therefore must be modified later
Data for a typical car – don’t use this, find data for your own car or family car
Car length - wheelbase 260 cm, body 400 cm
Car width - wheelbase 175 cm, body 195 cm
Kerb to Kerb turning diameter 1100 cm – use to calculate the Wall-to-Wall turning radius
Car Parking summary March 2022.docx 08/03/22
ENG2012 MATHEMATICAL MODELLING
PARKING A CAR
1. The original problem was concerned with the simple case of a car as shown in the
diagram, allowing us to find a rough answer to the problem – i.e. the overall minimum
length of the gap (L+Lr+D), converting to the wall-to-wall radius if only the kerb-to-
kerb radius is available. Eliminate x to find an expression for D.
2. For your own car or your family car, find the data for these parameters and calculate
the minimum gap from which a car can be driven out. Check that it’s a sensible value,
and use an appropriate level of precision for calculations and results.
3. Try to complete the ‘chart’ of Assumptions and Simplifications for Car, Driver,
Environment.
4. Simple changes to the basic problem can include varying the width of the car in front,
or making that a kerb instead of a car/wall.
5. Once the car has just cleared the corner of the car in front, what would the driver do
next? The aim is to bring the car to a parallel position (this would be the starting
position for backing into the space if we were doing the problem that way round). Can
you draw suitable diagrams for this, labelled with R and other parameters?
You should record your work on the worksheet and submit by midnight on Monday 14th
March via Canvas.
This piece of work will not be marked with a score, but will be counted as evidence of
attendance and engagement. If the final assignment is on this topic, then your preliminary
work at this stage will be invaluable.