Circular Motion on a Bank Question

[avonrepus]

I have a general question about circular motion of a car on a
frictionless bank.

What would be the function of the path (what is the shape?)
of the car entering a bank with a velocity v, and is slipping upwards
because the v is too high to for mv^2/r = Nsinθ.

The initial velocity is going straight into the page (When the cross section view of the bank is made)
The path is to be studied from when it enters the bank to where it is moving
in circular motion.

Please describe the steps needed to solve this problem.

 

[member 553083]

The path of the car is determined by three main factors: the initial velocity, the radius of curvature, and the coefficient of friction. Since the bank is assumed to be frictionless, the coefficient of friction is zero and the car will travel in a straight line until it reaches the point where the centripetal acceleration (i.e. mv^2/r) equals the normal force (Nsinθ). At this point, the car's path will bend sharply and begin to follow a circular path around the bank. To solve for the path more precisely, we can use the equation of motion for uniform circular motion, which states that the radial acceleration (ar), the tangential acceleration (at), and the angular velocity (ω) are related by the equation ar = ω^2r + at. The angular velocity can be calculated from the initial velocity and the radius of curvature, and then the radial and tangential accelerations can be calculated from the angular velocity. Finally, the equation of motion can be solved to determine the path of the car.