# Math and Science Solutions for Businesses/Turning circle for van

Question
i am considering purchase of van, and need to know how to determine if they will be able to make the turn from a narrow alleyway to the area behind my building.

so if i have a alleyway thats x feet wide, and runs along a building, and the alleyway runs x additional feet past the building before it terminates at a brick wall, how do i know if i will have enough room in that additional space to turn the vans (to the left as it happens) to access the space behind the building?

thanks much!

turning van
Hi Abe,

This question is actually a little more tricky than I first thought. The key consideration is the calculation of the so-called "offtracking" distance, which is how much the back wheels tend to follow inside the circle described by the (steering) front wheels. The attached figure shows a schematic of a left turning vehicle. The horizontal and vertical lines represent the walls of the alleyways you are trying to maneuver in. The distances Xmin and Ymin are the minimum distances from the walls you need in order to turn your van given its width, length and turning radius. The point where they meet corresponds to the corner of the inside walls of the alleyways (I think my Xmin corresponds to your first "x" and Ymin to your 2nd "x").

Define

W = van width
L = length

Then, distances along the radius from the perimeter of the turning circle (near outside corner) are

OT = R - sqrt(R^2 - L^2)
CL = [sqrt(2)-1)R
D = CL + W/2 + OT

Xmin = D/sqrt(2) + W/2

Ymin = D/sqrt(2) + sqrt(2)W/2.

I'm afraid this is a little cryptic and I wish I had a better schematic (my drawing package is not great). The sqrt(2) in the denominators take into account going from distances along the diagonal to the x and y axes. The extra sqrt(2) for Ymin takes into account the distance the width of the van makes at a 45 degree angle to the vertical wall as the van turns (so you don't scrape the wall).

For

W = 6 ft
L = 12 ft
R = 18 ft (note: this is a radius = half the diameter of the turning circle)

I get

Xmin = 13.6 ft
Ymin = 14.8 ft.

These distances are the minimum in each direction. If either alleyway is wider than these mins, then you are fine in that direction and only have to worry about the other (if both are wider, then no problem). Note that this derivation assumes a perfect driver and zero tolerances, so adjust accordingly.

If you are interested, I could provide a more detailed schematic (probably hand drawn). Good question and good luck!
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks for the detailed response. I appreaciate the offer for followup drawing, and will run some numbers and see. Again, thanks for your help.

Math and Science Solutions for Businesses

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#### Randy Patton

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Questions regarding application of mathematical techniques and knowledge of physics and engineering principles to product and services design, optimization, prediction, feasibility and implementation. Examples include sales and product performance projections based on math/physics models in addition to standard regression; practical and cost effective sensor design and component configuration; optimal resource allocation using common tools (eg., MS Office); advanced data analysis techniques and implementation; simulation and "what if" analysis; and innovative applications of remote sensing.

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26 years as professional physical scientist and project manager for elite research company providing academic quality basic and applied research for government and defense industry clients (currently retired). Projects I have been involved in include: - Notional sensor performance predictions for detecting underwater phenomena - Designing and testing guidance algorithms for multi-component system - Statistical analysis of ship tracking data and development of anomaly detector - Deployed vibration sensors in Arctic ice floes; analysis of data - Developed and tested ocean optical instrument to measure particles - Field testing of protoype sonar system - Analysis of synthetic aperture radar system data for ocean surface measurements - Redesigned dust shelters for greeters at Burning Man Festival Project management with responsibility for allocation and monitoriing of staff and equipment resources.

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“A Numerical Model for Low-Frequency Equatorial Dynamics” (with Mark A. Cane), J. of Phys. Oceanogr., 14, No. 12, pp. 18531863, December 1984.

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MIT, MS Physical Oceanography, 1981 UC Berkeley, BS Applied Math, 1976

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