# Designing a Banked, Circular Highway Exit | Help with Homework

• Yellowkies_3275
In summary, the task is to design a banked, circular, clockwise highway exit on 5000 sq. meters of land. The average stretch before entering a highway exit is typically 300 m and the highway speed limits are 100 km/h but can reach 120 km/h. Vehicles can range in weight from 100 kg to 100,000 kg. The goal is to determine the speed limit and angle of the bank for the curve, taking into account different road conditions such as rain and snow. The equations used for this problem are v= √g*r*tanθ and V=√[tanθ+μ/1-μtanθ]*r*g. The weight of vehicles is not a factor in these
Yellowkies_3275

## Homework Statement

The task is to design a highway. I did a similar project last year but I have completely forgotten and have no clue what to do.

You have to design a banked, circular, clockwise highway exit. you knwo you have 5000 sq. meters. of land to build it on..you also know that the average stretch before entering a highway exit is typically 300 m. you know the highways speed limits are 100 km/h but can reach 120 km/h (27.777 m/s and 33.333 m/s). you also know that vehicles can range in weight from 100 kg to 100,000 kg.

We have to determine the speed limit for the curve as well as determine the angle of the bank on the curve. and aslo take into account different road conditions such as rain and snow.

I've been looking at the problem for hours and tried some stuff but I only got ridiculous numbers because i used the highways speed limits because I don't know what the speed limit of the curve should be. I don't know what to do

## Homework Equations

I don't even know

I think v= √g*r*tanθ

## The Attempt at a Solution

[/B]
I have none for you right nowplease give me some guidance (I shouldn't have taken physics I'm so helpless)

Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.

where
haruspex said:
Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.
do i get this information from. it does not come with the lab? is it appropriate to search up
i know what forces act but i still can't see what to do both without a defined velocity and without an agle

haruspex said:
Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.
I also am curious, in the equations i know (V= √g*r*tanθ) (V=√[tanθ+μ/1-μtanθ]*r*g) weight isn't a factor, so why am i given the weight of vehicles?

haruspex said:
Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.
okay i think i have something that works could you just check if these numbers sound okay?

so knowing the radius would be 35 sq meters I tested out various angles because I really wasn't sure how to find a maximum logical one without just guessing and checking...because if you wanted you could make a 70 degree bank so that the car could continue to go 120 km/h even on the curve...but that's a bit extreme... so like many things were possible I think so I just tried to use common sense to figure out what was most reasonable...I ended up deciding to use a 45 degree angle for the curve, a bit steep but when you check the amount people would have to slow down to from the 100~120 km/h highway to drive the 45 degree curve the velocity still seems reasonable...fast maybe but efficient and still safe

r= 35 sq meters
θ=45°
V= on a frictionless surface (ice on road or something) 18.52 m/s or 66.667 km/h. Then on more regular conditions speeds of upto 20% increase can be expected so my max speed limit is 80 km/h (22.222 m/s) which would allow you to travel the curve of 45 degrees with a μ of 0.18

i still have no clue what to do with the info about the weight of vehivles traveling (range from 100 kg to 100,000 kg) logically in the real world i can understand why that info would be important you don't want any trucks tipping over, but nowhere in these banked curve equations is mass mentioned, mass gets mentioned seperatly in x or y direction but it cancels out so where is it relevent?

Yellowkies_3275 said:
if you wanted you could make a 70 degree bank
No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.

Yellowkies_3275
haruspex said:
No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.
How do i find the minimum guaranteed friction coefficient?

haruspex said:
No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.
isnt the minimum coefficient of friction let's say for a slippery ice covered road zero? so with 0 friction i would have to rely on v= √g*r*tanθ for which i don't know velocity or theta

on a reguar road without velocity or theta, where i have to judge for myself what is best how would i come up with friction, if my previous friction wasn't correct?

Perhaps I missed it but how do you know the radius is 35m?

Can you check the OP contains the problem statement word for word or that you have been told to make your own assumptions?

For example what shape is the 5000sqm? Was a diagram provided?

Yellowkies_3275 said:
isnt the minimum coefficient of friction let's say for a slippery ice covered road zero? so with 0 friction i would have to rely on v= √g*r*tanθ for which i don't know velocity or theta
The concern is that a vehicle going around the curve dead-slow over glare ice could slide down off the roadway to the inside. The relevant parameter is the coefficient of friction on glare ice. You should not assume that this coefficient is zero -- that would lead to your designing the curve with no bank angle at all.

You are the designer. You choose v, you choose r, you choose theta. You want choose them so that certain constraints are met. For instance, you might want to optimize for max permitted v by modifying road width, curve radius and bank under the constraints that land use not exceed 5000 m^2, and that cars neither slide off the inside of the roadway under glare ice conditions nor off the outside of the roadway in the wet at the posted speed limit.

You need at least some bank angle so that rain drains from the roadway. https://en.wikipedia.org/wiki/Drainage_gradient. You can also find guidance in https://en.wikipedia.org/wiki/Interstate_Highway_standards.

I found this link with various coefficients of friction. http://www.iaeng.org/publication/WCE2011/WCE2011_pp2381-2384.pdf

One approach that could be used would be to make the surface of the ramp concave upward with an inner shoulder at a lower bank angle than the traveled portion of the roadway -- if cars slide off inward, they only slide onto the shoulder.

## 1. What is the purpose of designing a banked, circular highway exit?

The purpose of designing a banked, circular highway exit is to ensure safe and smooth transition of vehicles from the high-speed highway to a lower speed exit ramp. The banked curve allows vehicles to maintain their stability and reduce the risk of accidents.

## 2. How is the banked, circular highway exit designed?

The banked, circular highway exit is designed by taking into consideration various factors such as the speed limit, radius of the curve, and the type of vehicles using the exit. Engineers use mathematical equations to calculate the appropriate banking angle and slope for the exit ramp.

## 3. What are the benefits of a banked, circular highway exit?

Some of the benefits of a banked, circular highway exit include improved safety for drivers, reduced risk of accidents, and smoother transition onto the exit ramp. It also allows for higher speed limits on the exit ramp, which can improve traffic flow.

## 4. What are the challenges faced in designing a banked, circular highway exit?

One of the main challenges in designing a banked, circular highway exit is finding the right balance between the banking angle and the radius of the curve. If the angle is too steep or the curve is too tight, it can lead to accidents. Another challenge is designing the exit ramp to accommodate different types of vehicles, such as cars, trucks, and motorcycles.

## 5. What are some key considerations to keep in mind when designing a banked, circular highway exit?

When designing a banked, circular highway exit, engineers must consider factors such as the speed limit, traffic volume, and the type of vehicles using the exit. They must also take into account the terrain and weather conditions of the area. It is also important to regularly monitor and maintain the exit ramp to ensure its safety and functionality.

• Introductory Physics Homework Help
Replies
20
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
723
• Introductory Physics Homework Help
Replies
9
Views
2K
• Introductory Physics Homework Help
Replies
12
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
4K
• Introductory Physics Homework Help
Replies
11
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
2K