What Is the Maximum Constant Speed of a Motorcyclist on a Curved Path?

[aaronfue]

Homework Statement

A motorcyclist travels around a curved path that has a radius of 450 ft. While traveling around the curved path, the motorcyclist increases speed by 1.10[itex]\frac{ft}{s}[/itex]. Determine the maximum constant speed of the motorcyclist when the maximum acceleration is 7.00 [itex]\frac{ft}{s^2}[/itex]

Homework Equations

a = √(a_t)²+(a_n)²

a_t=[itex]\dot{v}[/itex]
a_n = [itex]\frac{v^2}{ρ}[/itex]

The Attempt at a Solution

I've already solved for the speed at a given acceleration, and the magnitude of the acceleration at a given speed.

But this part is a little confusing for me. I am thinking that I have to use the equation a = √(a_t)²+(a_n)² and set a = 7.00 [itex]\frac{ft}{s^2}[/itex] and then I can solve for a_n...then solve for v in the equation a_n = [itex]\frac{v^2}{ρ}[/itex], where ρ = 450 ft? But I'm not sure. I will give it a try, though.

If anyone can weigh in on my approach, I would greatly appreciate it as always!

 

[KataKoniK]

Hi there! It looks like you're on the right track with your approach. To find the maximum constant speed of the motorcyclist, you need to find the point where the acceleration is at its maximum value, which is 7.00 \frac{ft}{s^2} in this case. This means that both the tangential acceleration (at) and the normal acceleration (an) are at their maximum values.

To find the maximum tangential acceleration, you can use the equation at=\dot{v}, where \dot{v} is the rate of change of speed. In this case, the motorcyclist is increasing speed by 1.10 \frac{ft}{s}, so the tangential acceleration is 1.10 \frac{ft}{s^2}.

To find the maximum normal acceleration, you can use the equation an = \frac{v^2}{ρ}, where v is the speed and ρ is the radius of the curved path. Since you're looking for the maximum speed, you can set the maximum normal acceleration to be equal to 7.00 \frac{ft}{s^2} and solve for v.

So your steps would be:
1. Use the equation at=\dot{v} to find the maximum tangential acceleration.
2. Use the equation an = \frac{v^2}{ρ} and set it equal to 7.00 \frac{ft}{s^2} to find the maximum speed.
3. Substitute the maximum speed into the equation an = \frac{v^2}{ρ} to find the corresponding maximum normal acceleration.

I hope this helps! Let me know if you have any other questions.