---
title: "A car of mass \\(m\\) travels at a constant speed \\(v\\) along a circular track of radius \\(r\\). The track is banked at an angle \\(\\theta\\) relative to the horizontal, as shown in Figure 1. The coefficient of static friction between the car’s tires and the track is \\(\\mu_s\\). The car is traveling at the maximum possible speed \\(v_{max}\\) such that it does not slide up the banked track."
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url: "https://nerd-notes.com/ubq/114491/"
date_modified: "2026-07-03T04:38:47+00:00"
---

# A car of mass \(m\) travels at a constant speed \(v\) along a circular track of radius \(r\). The track is banked at an angle \(\theta\) relative to the horizontal, as shown in Figure 1. The coefficient of static friction between the car’s tires and the track is \(\mu_s\). The car is traveling at the maximum possible speed \(v_{max}\) such that it does not slide up the banked track.

A car of mass \(m\) travels at a constant speed \(v\) along a circular track of radius \(r\). The track is banked at an angle \(\theta\) relative to the horizontal, as shown in Figure 1. The coefficient of static friction between the car's tires and the track is \(\mu_s\). The car is traveling at the maximum possible speed \(v_{max}\) such that it does not slide up the banked track.

![A car, represented by a rectangular box, rests on a cross-section of a track banked at an angle \(\theta\) relative to the horizontal. The incline slopes upward from left to right. A horizontal dashed line extends from the car to the left, terminating at a vertical dashed line. The radius \(r\) is labeled along this horizontal dashed line, indicating the center of the circular track is to the left.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1783053526-BDkO6o.jpg)

**Part a)** On the dot below, which represents the car, **draw** and **label** the forces (not components) that are exerted on the car when it is traveling at \(v_{max}\). Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

**Part b)** Student A makes the following claim: "If the car travels faster than \(v_{max}\), a centrifugal force pushes the car radially outward, causing it to slide up the incline." **Identify** the flaw in Student A's reasoning. Then, in a clear, coherent, paragraph-length response, **explain** the actual reason the car would slide up the incline if it exceeded \(v_{max}\). Your explanation must rely on physical principles and reference the specific forces acting on the car.

**Part c)** **Derive** an expression for the maximum speed \(v_{max}\) of the car. Express your answer in terms of \(m\), \(r\), \(\theta\), \(\mu_s\), and fundamental constants as appropriate.

**Part d)** Student B makes the following claim: "If the track is covered in ice such that the friction between the tires and the track is negligible (\(\mu_s = 0\)), the car must travel at exactly \(v = 0\) to avoid sliding down the incline."


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/114491/*