Calculating the Speedbump Radius to Stop a Car: A Homework Challenge

[DannyTatas]

Homework Statement

A car goes over a speedbump, which has the cross-section of a cylinder of radius R embedded in the roadway. If you want a car driving with a speed Vo to be impeded, how large must R be?

I'm very confused about this problem because we have not discussed the topic in class and I have a terrible book. I want to say that you use the centripetal force equation, but that is for when velocity is constant? I'm not sure...

Knowns:
V initial = Vo
V final = 0
mass = m
Radius = height of the speed bump

Unknowns:
Radius

Homework Equations

F = m(v^2)/R

The Attempt at a Solution

Again, I am having trouble and do not know where to begin. I want to learn how to do this; not get a quick answer. Any and all help will be greatly appreciated.

 

[Spinnor]

We need two numbers to better define the speed bump, the radius R and the height of the section of a cylinder. Is it half a cylinder, a quarter? I hope you understand.

As for impeding the car you need to be more specific. A car with a very stiff suspension will handle the speed bump differently then a car with a very soft suspension. I think we need more information.

 

[DannyTatas]

This is just how my professor operates. He makes us solve variables with other variables rather than numeric values.

 

[rcgldr]

A speed bump won't slow down a car much (assuming it doesn't damage the car's tires, wheels or suspension). Perhaps the goal is to find R so that v0² / R translates into 1 g of acceleration, but even that would not slow down a car much.

 

[Nickie]

As a scientist, it is important to first understand the problem and the given information before attempting to solve it. In this case, we are given the speed of the car (Vo) and the cross-section of the speedbump (radius R). Our goal is to find the required radius R in order to impede the car's speed.

To begin, we can use the equation for centripetal force, which is F = m(v^2)/R. This equation relates the force (F) required to keep an object moving in a circular path with its mass (m), velocity (v), and radius of the circle (R). In this case, we can think of the speedbump as a circular path that the car must travel over.

We know that the final velocity of the car must be zero, as it comes to a stop after going over the speedbump. We also know the initial velocity (Vo) and the mass of the car (m). Therefore, we can rearrange the centripetal force equation to solve for the radius R:

R = m(v^2)/F

We can use the force of friction (Ff) as the force acting on the car to slow it down. This force is dependent on the coefficient of friction (μ) between the car's tires and the speedbump. We can rewrite the equation as:

R = m(v^2)/μmg

Where g is the acceleration due to gravity. This equation gives us the required radius R in terms of the car's mass, velocity, and the coefficient of friction. We can use this equation to calculate the radius for different scenarios, such as different speeds or different coefficients of friction, to determine the optimal size for the speedbump.

In conclusion, to calculate the required radius of a speedbump to impede a car's speed, we can use the equation R = m(v^2)/μmg, where m is the car's mass, v is its initial velocity, μ is the coefficient of friction, and g is the acceleration due to gravity. It is important to understand the concept of centripetal force and its application in this problem before attempting to solve it.