Problem 37 - Calculus of a Single Variable

Question

Roadway Design Cars on a certain roadway travel on a circular arc of radius \(r .\) In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude \(\theta\) from the horizontal (see figure). The banking angle must satisfy the equation \(r g \tan \theta=v^{2},\) where \(v\) is the velocity of the cars and \(g=32\) feet per second per second is the acceleration due to gravity. Find the relationship between the related rates \(d v / d t\) and \(d \theta / d t\) .

Step-by-Step Solution

Answer

The relationship between the related rates \(d v / d t\) and \(d \theta / d t\) is \( d \theta / dt = 2 \cdot dv/dt /v \)

Step 1: Understand the Given Formula

The given formula is \(r g \tan \theta = v^2\). The radius \(r\) and acceleration due to gravity \(g\) are constants and \(\theta\) is the angle that the roadway is banked at which yields the velocity squared (\(v^2\)).

Step 2: Find the Derivative of the Given Formula

Differentiating the given formula with respect to time \(t\) gives us \(r g \sec^{2} \theta \cdot d \theta / dt = 2v \cdot dv/dt\). It's crucial to note that \(\sec^2 \theta = 1/ \cos^2 \theta\) and \(2v\) is from the chain rule of differentiation \(d(v^2)/dt = 2v \cdot dv/dt\).

Step 3: Simplify the Differentiated Formula

Rearranging equation, \(d \theta / dt = 2v \cdot dv/dt /(r g \sec^{2} \theta)\), by identifying \(\sec^2 \theta\) with \(1+\tan^2 \theta\) and exploiting the original formula, we obtain \(d \theta / dt = 2v \cdot dv/dt /(v^2)\) which further simplifies to \( d \theta / dt = 2 \cdot dv/dt /v \)

Key Concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned with the study of how functions change when their inputs change. It's the mathematical way of finding the rate at which one quantity changes with respect to another. In solving problems involving related rates, like the rate of change of the banking angle of a roadway, you often work with derivatives, which are the fundamental tool in differential calculus.

The derivative of a function at a point is the slope of the tangent line to the function at that point. It represents the instantaneous rate of change of the function with respect to a given variable. In the context of the roadway design problem, the process of finding the related rates involves taking the derivative of the given formula with respect to time using the chain rule to establish a relationship between the rates of change of velocity and the banking angle.

Tangential Velocity

Tangential velocity refers to the velocity of an object moving along a circular path. It is always directed at a tangent to the circular path, hence the name. The magnitude of the tangential velocity is equal to the rate of change of the angular displacement over time. In the case of a car turning on a banked roadway, the tangential velocity is the speed at which the car travels along the arc of the roadway. This concept is central to understanding how the banking angle is related to the velocity of the cars since the centrifugal force acting on the vehicles is dependent on their tangential velocity.

Banked Roadway Physics

The physics of a banked roadway revolves around creating a safer and more efficient path for vehicles when turning. By banking the turn, the roadway inclines at an angle such that a component of the gravitational force counteracts the centrifugal force that tends to push vehicles outwards in a turn. This allows for higher speeds and/or reduced dependency on friction.

In a banked turn, the inward centripetal force is provided in part by gravity rather than by friction alone. The angle of banking \theta plays a critical role in balancing these forces. The expression \tan \theta relates the force of gravity to the centrifugal force experienced by a car traveling at velocity \(v\) on a curve of radius \(r\). Understanding this relationship is key to solving problems regarding the rate of change of the banking angle as the velocity of cars changes.

Centrifugal Force

Centrifugal force is a fictitious or apparent force that appears to act on a body moving in a circular path and directs it away from the center around which the body is moving. This force arises from the inertia of the object, which tends to follow a straight-line path. As a result, there must be a counteracting force to keep the object in its circular path – the centripetal force.

On a banked roadway, the centrifugal force must be balanced by the centripetal force to keep a car moving in a circular turn without skidding. This balance is what informs the proper banking angle for the road, ensuring that even if friction were negligible, a car could still navigate the turn. The centrifugal force is directly related to the square of the tangential velocity and the mass of the vehicle and is inversely proportional to the radius of the turn.