consider a racecar with some mass M going at some speed V, on a road with some width W. (the racecar has a negligible size). the racecar encounters a curve that follows an unbanked curve that can be defined as a quarter circle with a radius of R. the driver of the racecar, being in the midst of race, wants to maintain the maximum speed while passing through the curve. define the path that the car must take to maintain the maximum possible speed.
according to the racing games i've played, the answer to this problem is to travel the circle with the largest possible radius
how does one prove that?
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Anonymous2006-11-18 18:57
depends on the handling of the vehicle, tire friction etc.
youd look at the max centripedal force the car can handle without slippage, and expand the path of the curve until you got it. thats why a bank allows you to maintain higher speeds in tighter turns, it provides support against the force acting on the vehicle away from the center of the turn-circle.
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Anonymous2006-11-18 18:58
Centripetal force = mv^2/r
Friction can only provide so much force before your car starts slipping off. The larger your radius, the higher your velocity can be before that happens. By banking curves, gravity can supplement the friction providing the centripetal force.
Note that maximum speed doesn't necessarily mean you win the race; you have to travel more distance the larger your radius.
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Anonymous2006-11-18 19:11
Imagine this, the curve goes to the right.
The car must go from the left hand side of the road to the inner corner of the curve.
Once it is past the tangent of that curve in relation to it's direction it must take a curve which goes as fast as possible till it can go in a straight line again without slippage. This would mean it would be very close to the outer corner of ther curve. In fact I'll draw a picture.
Bear in mind this rule is an approximation. If the curve the race car was on slowly curved around for a mile or so the race car would want to take the tightest curve possible when it has to turn. For curves only 20-100 metres or so this rule applies.