# Car on a banket - maximum speed formula derivation

• timarli
In summary, the conversation discusses the topic of finding the maximum speed of a car negotiating a bend, taking into consideration factors such as the slope of the road, coefficient of friction, weight of the car, and radius of the bend. The individual mentions attempting to use a formula involving the weight of the car and the parallel and perpendicular components of the weight, but realizes that according to the book, there is a second component of the centripetal force that must be considered. They also share a link to a paper discussing this topic. Ultimately, they come to the conclusion that the centripetal force acts horizontally, not parallel to the surface, and thank another individual for clearing up their misconception.
timarli
Hi,

This is not related with a specific homework question. I was studying this topic and have noticed that I didn't understand some bits.The car is negotiating a bend with a speed of V.
The slope of the banket is θ
The coefficient of friction is η
Weight of the car if mg
Radius of the bend is R

To find the maximum speed I broke down the weight into two components, parallel and perpendicular to the road surface. So the parallel component + η times perpendicular component should give me the component of the centripetal force that's parallel to the surface. [m*g*cosθ*η + m*g*sinθ] = (m*v^2)*cosθ

But looks like I am wrong because according to the book the force that's perpendicular to the surface is not only from gravity but has a second component coming from 'centripetal force'? This is the actual formula : [(m*g*cosθ + m*v^2*cosθ/R)*η + m*g*sinθ] = (m*v^2)*cosθ

I really don't understand this one because the "centripetal force" itself is the resultant of weight and force applied by the surface on the car. There is nothing else that can cause this central acceleration. So adding a component of the centripetal force to the above formula to find the 'centripetal force itself does not make sense at all :S

Last edited:
the centripetal force acts in the direction of centre of the circle in which the the object is moving.in this case,it would be the center of the horizontal circle on which the car moves. so,the centripetal force is not acting parallel to the surface,but instead,horizontally.

Thanks a lot utkarsh5. Like you said I was using different axis' where I was simplifying horizontally without realizing that I was introducing a new component.

Thanks again :)

## 1. How do you calculate the maximum speed of a car on a banked track?

The maximum speed of a car on a banked track can be calculated using the following formula: vmax = √(rgtanθ), where vmax is the maximum speed, r is the radius of the track, g is the acceleration due to gravity, and θ is the angle of banking.

## 2. What is the importance of a banked track in increasing a car's maximum speed?

A banked track allows a car to maintain a higher speed while turning because the force of friction is directed towards the center of the turn, providing additional centripetal force. This reduces the reliance on the car's tires for turning, allowing the car to maintain a higher speed.

## 3. How does the radius of the track affect the maximum speed of a car on a banked track?

The radius of the track is directly proportional to the maximum speed of a car on a banked track. This means that a larger radius will result in a higher maximum speed, while a smaller radius will result in a lower maximum speed. This is because a larger radius allows for a smoother turn and requires less centripetal force.

## 4. Why is the acceleration due to gravity included in the maximum speed formula for a car on a banked track?

The acceleration due to gravity plays a crucial role in determining the maximum speed of a car on a banked track. This is because the force of gravity helps to maintain the car's motion towards the center of the turn, providing additional centripetal force and allowing the car to maintain a higher speed.

## 5. What happens if the angle of banking on a track is too steep?

If the angle of banking on a track is too steep, it can lead to a decrease in the maximum speed of a car. This is because a steeper angle of banking requires a higher centripetal force to maintain the car's motion towards the center of the turn. If this force is not provided, the car may slide off the track and lose speed.

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