Solving Mechanics Q: Who Wins Car A or B at Turn?

[nlsherrill]

Homework Statement

Two racing cars apprach a turn. Car A is in the inside lane and car B is in the outside lane. The two cars travel through the turn at a constant speed. Just before the turn the two cars are side by side.

a)If the friction coefficient is infinite, which car will be ahead at the end of the turn?

b)Consider now the friction coefficient for the tires on the asphalt. Which car is ahead after the turn?

Homework Equations

The Attempt at a Solution

First off this may seem like a intro physics question, and it kind of is, but its from my classical mechanics class and the semester just started, so this is kind of a "review" of elementary Newtonian mechanics.

For part a), I said that Car A would be ahead after the turn, because it can travel the same speed as Car B, and has less distance to travel. I think this is right.

For part b, I am not sure. We asked our professor in class if the angle of the turn or the width of the lanes had anything to do with it and he said "no, there is a way to show which car would end up ahead".

I am not sure how to show this. Wouldn't car A still be ahead? It says that the two cars travel through the turn at constant speed, and that they are side by side. I guess this doesn't imply that they are traveling at the same constant speed through the turn. I don't know how slow A would have to go on asphalt compared to B due to the tighter turn it would have to make. Intuitively, A would have to go slower, but at the same time has less distance to travel, and B could go faster with more distance to travel. How is there a way to know with only the information given?

 

[sgd37]

for a) you are correct for b) car B would come out of the turn first since acceleration due to change in direction (all other things being equal) is proportional to R i.e the radius of the circle

Just look up circular motion

 

[nlsherrill]

for a) you are correct for b) car B would come out of the turn first since acceleration due to change in direction (all other things being equal) is proportional to R i.e the radius of the circle

Just look up circular motion

So is this assuming that the centripetal force is the force of friction holding the car's tires to the road, and that the larger the radius of curvature the larger the amount of friction acting on the tires, thus the car on the outside lane can go faster?

Is this because centripetal force=mv^2/r, so the velocity will grow faster by a little increment(10 m/s vs. 12 m/s) due to the squared term versus the radius of the circle which isn't squared?

 

[sgd37]

yes the centripetal force would be equal to the force of friction otherwise the car would go flying off the road. What you mention are all general concepts of fixed radius circular motion

 

[paranormal]

As a scientist, it is important to use the information given to make logical and evidence-based conclusions. In this scenario, we can use Newton's laws of motion to determine which car will be ahead at the end of the turn.

For part a), if the friction coefficient is infinite, it means that there is no limit to the amount of friction between the tires and the road. In this case, both cars will be able to maintain their constant speed and will not experience any loss of momentum. Since car A is in the inside lane, it will have a smaller radius to travel around the turn compared to car B in the outside lane. This means that car A will cover a shorter distance in the same amount of time, and will therefore be ahead at the end of the turn.

For part b), we need to consider the friction coefficient for the tires on the asphalt. This coefficient determines the amount of friction between the tires and the road, and will affect the speed at which the cars can travel through the turn. If the friction coefficient is high, both cars will be able to maintain a higher speed through the turn without slipping. However, since car A has a smaller radius to travel around the turn, it will require a higher centripetal force to maintain its speed. This means that car A will have to slow down more than car B, and therefore car B will be ahead at the end of the turn.

In conclusion, the answer to which car will be ahead at the end of the turn depends on the friction coefficient for the tires on the asphalt. If the friction coefficient is infinite, car A will be ahead. However, if the friction coefficient is finite, car B will be ahead due to its larger radius and ability to maintain a higher speed through the turn. It is important to consider all factors, such as friction, radius, and speed, when making predictions in mechanics.