Cloverleaf highway interchange - determine car acceleration

In summary, the curved portion of a cloverleaf highway interchange is defined by the equation ##R^2=b^2sin2θ##, with a car traveling along the curve at a constant speed of v0. To determine the car's acceleration at point A, the expression for ##\ddot R## should be re-evaluated to include a ##\ddot \theta## term and correct for the missing ##\dot\theta^2## term. Dimensional analysis can also be used to check the correctness of the equations. Additionally, when taking the second derivative, it must be remembered that both ##\cos(2\theta)## and ##(\sin(2\theta))^{-1/2}##
  • #1
Alexanddros81
177
4

Homework Statement


13.34 The curved portion of the cloverleaf highway interchange is defined by
##R^2=b^2sin2θ##, 0<=θ<=90deg. If a car travels along the curve at the constant speed v0,
determine its acceleration at A

Fig13_34.jpg


Homework Equations

The Attempt at a Solution



Pytels_Dynamics071.jpg

[/B]
Is this wrong?
 
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  • #2
Check your expression for ##\ddot{R}##. There should be a ##\ddot{\theta}## in it.​
 
  • #3
and also you're missing a ##\dot \theta^2##.
 
  • #4
kuruman said:
Check your expression for ¨RR¨\ddot{R}. There should be a ¨θθ¨\ddot{\theta} in it.

Is the following correct:

Pytels_Dynamics072.jpg


Marc Rindermann said:
and also you're missing a ˙θ2θ˙2\dot \theta^2
I cannot see where I am missing the ##\dot θ^2##
 
  • #5
first of all you can check whether your equations are correct if you do a dimensional analysis.
Look at ##\ddot R = 2b\dot\theta##

##[\ddot R] = ms^{-2}##
##[2b\dot\theta] = ms^{-1}##

So you see something is not quite right.

When you take the 2nd derivative of ##\dot R = b\cos(2\theta)(\sin(2\theta))^{-1/2}\dot\theta## you need to remember that the derivative of both ##\cos(2\theta)## and ##(\sin(2\theta))^{-1/2}## produces another ##\dot\theta##.
 
  • #6
Pytels_Dynamics088.jpg

Pytels_Dynamics093.jpg

Pytels_Dynamics089.jpg


Can you check this?
 

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1. How does a cloverleaf highway interchange work?

A cloverleaf highway interchange is designed to allow two highways to intersect without any traffic signals or stop signs. The roads are connected by a series of curved ramps and loops, resembling a four-leaf clover when viewed from above. This design allows for continuous traffic flow in all directions.

2. What is the purpose of determining car acceleration in a cloverleaf interchange?

In a cloverleaf interchange, cars must navigate the curved ramps and loops at varying speeds. Determining car acceleration helps engineers understand how well the interchange is functioning and if any adjustments need to be made to improve traffic flow.

3. How do scientists determine car acceleration in a cloverleaf interchange?

Scientists use mathematical equations and formulas to determine car acceleration. They measure the distance traveled and the time it takes for a car to travel through the interchange, and then use this data to calculate the car's acceleration.

4. What factors affect car acceleration in a cloverleaf interchange?

There are several factors that can affect car acceleration in a cloverleaf interchange, including the angle and radius of the curves, the grade of the ramps, and the speed limit. Weather conditions and the type of vehicle being driven can also impact acceleration.

5. How can car acceleration be improved in a cloverleaf interchange?

To improve car acceleration in a cloverleaf interchange, engineers can make adjustments to the design, such as increasing the radius of the curves or reducing the grade of the ramps. Increasing the speed limit or implementing traffic control measures, such as metered ramps, can also help improve acceleration.

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