Question

Suppose that a car with a door open at right angles \(\left(\theta=90^{\circ}\right)\) starts up and accelerates at a constant rate \(a=1 \mathrm{mph} / \mathrm{sec} .\) The differential equation for \(\theta(t)\) is \(\vec{\theta}=-A \sin \theta\) where \(A=3 a / 2 w\) for a uniform door of width \(w\). If \(w=3.5 \mathrm{ft}\), find how long it takes for the door to close.

Short answer

The door theoretically never closes due to infinite time differentiation.

Step-by-step solution

– Understand the given variables

Given, Acceleration, \(a = 1 \text{ mph/sec}\).Width of the door, \(w = 3.5 \text{ ft}\). The differential equation provided is \(\theta'(t) = -A \times \text{sin}(\theta)\). \(A\) is given by \(A = \frac{3a}{2w}\).

Calculate the constant A

Using the given values, \(A = \frac{3a}{2w}\).Plug in the values \(a = 1 \text{ mph/sec}\) and \(w = 3.5 \text{ ft}\):\[A = \frac{3 \times 1}{2 \times 3.5} = \frac{3}{7} \text{ sec}^{-1}\]

Solve the differential equation

The differential equation is \[\theta'(t) = -\frac{3}{7} \times \text{sin}(\theta)\]. Separate variables and integrate to find \(t\):\[\int \frac{d\theta}{\text{sin}(\theta)} = -\frac{3}{7} \int dt\].This simplifies to: \[-\ln|\text{tan}(\theta/2)| = -\frac{3}{7}t + C\].

Apply initial condition

Use the initial condition that \(\theta = 90^\text{°}\) or \(\frac{\pi}{2}\) when \(t = 0\).Since \(\text{tan}(\frac{\pi}{4}) = 1\), \[-\ln(1) = -\frac{3}{7} \times 0 + C \rightarrow C = 0\].

Solve for the time to close the door

Set \(\theta = 0\) to find when the door closes and solve for \(t\):\[\ln|\text{tan}(0/2)| = -\frac{3}{7} t\].Since \(\ln|0|\) is undefined, we look at the limit,as \(\theta \rightarrow 0\), \(\text{tan}(\theta/2)\rightarrow 0\). Therefore, we solve\[-\frac{3}{7}t = -\infty \Rightarrow t = 0\].

Key concepts

acceleration

In physics, acceleration is the rate at which the velocity of an object changes with time. It’s represented by the symbol \(a\). For the car in our problem, the acceleration is given as 1 mph/sec. This means every second, the car's speed increases by 1 mile per hour.
Understanding acceleration is crucial because it directly influences how forces interact with an object in motion. For our example, the car's door is affected by this accelerating force, which creates a need to analyze how it changes over time. The acceleration is used to determine the constant in the differential equation that we need to solve.

initial conditions

Initial conditions are the starting values that define the state of a system at the beginning of our observations. They are vital because they help solve differential equations by giving us specific information at \( t = 0 \).
In our car door problem, the initial condition is that the door is open at right angles, meaning \( \theta = 90^{\text{°}} \) or \( \pi/2 \) radians when \( t = 0 \). This initial condition is used to determine the constant of integration (often represented as \( C \)) in our solution. Using this, we substitute back into our equation to find the variable we are solving for—time until the door closes in this case.

variable separation

Variable separation is a method used to solve a differential equation by moving all terms involving one variable to one side of the equation and the terms involving the other variable to the opposite side.
In our exercise, we begin with the differential equation \( \theta'(t) = -A \sin(\theta) \). To separate variables, we re-arrange it as follows: \[ \int \frac{d\theta}{\sin(\theta)} = -A \int dt \] This re-arrangement makes each side of the equation dependent on only one variable. Once separated, we integrate both sides to solve for \( \theta \) as a function of \( t \). This approach is often used because it simplifies the process of solving complex differential equations.

integration

Integration is the process of finding the integral of a function, which can be understood as the reverse process of differentiation. In solving differential equations, integration plays a crucial role in finding the function that describes the system.
In our given problem, after separating the variables: \[ \int \frac{d\theta}{\sin(\theta)} = -A \int dt \] we need to integrate both sides. The integral on the left, \( \int \frac{d\theta}{\sin(\theta)} \) simplifies to \( -\ln|\tan(\theta/2)| \). The integral on the right, \( -A \int dt \), simplifies to \( -At + C \). Using initial conditions, we solve for the constant \( C \), and then we solve for \( t \) when \( \theta = 0 \), indicating the door has closed.
Integration allows us to move from a rate of change (differential equation) to a specific solution, giving us the required result.