91Ó°ÊÓ

A person is parachute jumping. During the time between when she leaps out of the plane and when she opens her chute, her altitude is given by an equation of the form \(y=b-c\left(t+k e^{-t / k}\right)\) where \(e\) is the base of natural logarithms, and \(b, c\), and \(k\) are constants. Because of air resistance, her velocity does not increase at a steady rate as it would for an object falling in vacuum. (a) What units would \(b, c\), and \(k\) have to have for the equation to make sense? (b) Find the person's velocity, \(v\), as a function of time. [You will need to use the chain rule, and the fact that \(\left.d\left(e^{x}\right) / d x=e^{x} .\right]\) (answer check available at lightandmatter.com) (c) Use your answer from part (b) to get an interpretation of the constant \(c\). [Hint: \(e^{-x}\) approaches zero for large values of \(\left.x .\right]\) (d) Find the person's acceleration, \(a\), as a function of time.(answer check available at lightandmatter.com) (e) Use your answer from part (d) to show that if she waits long enough to open her chute, her acceleration will become very small.

Step by step solution

01

Understanding Units for Constants

To ensure the equation \(y = b - c(t + ke^{-t/k})\) has consistent units, \(b\) must have units of length(e.g., meters), since \(y\) represents altitude. The term \(ct\) must also have units of length, making the unit of \(c\) length per time (e.g., meters per second) and \(t\) time (e.g., seconds). As a result, \(k\) must have units of time (e.g., seconds) to match with the units when arguing dimensional consistency within the negative exponential component.

02

Finding Velocity as a Function of Time

The velocity \(v(t)\) is the derivative of altitude \(y(t)\) with respect to time \(t\). Begin by finding the derivative of each term separately:\[ v(t) = \frac{dy}{dt} = -c \left(1 + \frac{d}{dt}(k e^{-t/k}) \right) \]Applying the chain rule to differentiate \(ke^{-t/k}\), we have:\[ \frac{d}{dt}(k e^{-t/k}) = k \cdot \left( -\frac{1}{k} \right)e^{-t/k} = - e^{-t/k} \]So, the velocity is:\[ v(t) = -c(1 - e^{-t/k}) \]

03

Interpreting the Constant 'c'

As \(t\) increases, \(e^{-t/k}\) approaches zero, thus making \(v(t)\) approach \(-c\). Therefore, the constant \(c\) can be interpreted as the terminal velocity of the object, which is the maximum velocity reached under constant air resistance when the effect of the exponential term becomes negligible.

04

Finding Acceleration as a Function of Time

Acceleration \(a(t)\) is the derivative of velocity \(v(t)\) with respect to time \(t\):\[ a(t) = \frac{dv}{dt} = c \frac{d}{dt}(e^{-t/k}) \]Using the chain rule again:\[ \frac{d}{dt}(e^{-t/k}) = -\frac{1}{k}e^{-t/k} \]Therefore, \[ a(t) = \frac{c}{k} e^{-t/k} \]

05

Analyzing Long-Term Acceleration

As time \(t\) goes to infinity, the exponential term \(e^{-t/k}\) approaches zero. Consequently, \(a(t) = \frac{c}{k} e^{-t/k}\) approaches zero as well. This illustrates that if the parachutist falls for a long duration, the acceleration due to gravity will become very small or essentially zero due to reaching terminal velocity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Terminal Velocity

In parachute jumping physics, a crucial concept to understand is terminal velocity. Imagine a skydiver free-falling from a plane. As they fall, they speed up until they reach a certain point where their speed no longer increases. This speed is called the terminal velocity. Terminal velocity happens when the force of gravity pulling the skydiver downward is perfectly balanced by the air resistance pushing up against them.
The skydiver's velocity at this stage stabilizes because both forces are equal and they no longer accelerate. This balanced state explains why the velocity doesn't change; there's no net force acting on the parachutist. In the provided exercise, the constant denoted by "c" is identified as the terminal velocity of the jumper, as the influence of the exponential term diminishes over time at this speed.

Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. When dealing with parachute jumping scenarios, calculating derivatives often involves using the chain rule. For this exercise, the skydiver's velocity is derived from the altitude function given in the problem.

• To differentiate the altitude equation, notice that one of the terms includes an exponential function which is composed of functions of time.
• By employing the chain rule, this derivative can be efficiently calculated. Suppose you have a function inside another function, differentiation through these layers is performed using this rule.

For instance, if you have a function in the form of \( f(g(x)) \), the derivative is the product of the outer function's derivative with respect to the inner function and the derivative of the inner function itself. This step-by-step differentiation leads us to understanding the velocity and acceleration of the parachutist accurately.

Air Resistance

Air resistance, also known as drag, is the opposing force air exerts against a moving object. In the context of parachute jumping, air resistance plays a critical role in controlling the descent.
Without air resistance, a skydiver would be in free fall with constant acceleration due to gravity. However, air resistance dramatically alters this state by exerting an upward force on the skydiver. This force increases with velocity until it balances out the gravitational force, leading to terminal velocity.

• The parachute itself is a device designed to maximize air resistance
• By deploying a parachute, the upward force of air resistance significantly intensifies and slows down the descent, adding a layer of control and safety.

In the provided problem, the term \( e^{-t/k} \) represents the influence of air resistance, as it affects both the velocity and acceleration calculations over time.

Dimensional Analysis

Dimensional analysis is a powerful tool in physics that ensures the consistency of equations by checking the dimensions of each term. Each physical quantity is assigned dimensions which relate to subcategories of measurement, like length, mass, and time.
In this problem, the altitude equation includes terms with different variables, each carrying specific dimensions. Ensuring dimensional consistency requires all terms concerning the same physical quantity, such as length or time, to have compatible units.

• For instance, in the parachuting equation, the constant \( b \) represents altitude, needing units of length (like meters).
• The term \( c(t + ke^{-t/k}) \) must also resolve to a length dimension to maintain the equation's validity.
• Hence, \( c \) corresponds to units of speed (length per time) and \( k \) must have units of time.

The dimensions must clearly align, providing a practical tool to validate the equation format and correctness when constructing or analyzing any physical equation.