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Chapter 5: Problem 1

If a skydiver jumps from an airplane, his velocity \(v\), in feet per second, starts at 0 and increases toward terminal velocity. An average-size man has a terminal velocity \(T\) of about 176 feet per second. The difference \(D=T-v\) is an exponential function of time. a. What is the initial value of \(D\) ? b. Two seconds into the fall, the velocity is \(54.75\) feet per second. Find an exponential formula for \(D\). c. Find a formula for \(v\). d. Express using functional notation the velocity 4 seconds into the fall, and then calculate that value.

Short Answer

Expert verified
a. Initial \(D\) is 176. b. \(D(t) = 176 e^{-kt}\). c. \(v(t) = 176 - 176 e^{-kt}\). d. \(v(4) = 176 - 176 e^{-4k}\) when calculated.

Step by step solution

01

Understanding Initial Value

At the start, the skydiver's velocity \(v\) is 0 feet per second. The terminal velocity \(T\) is given as 176 feet per second. Therefore, the initial value of \(D\), which is the difference \(T-v\), is \(D = 176 - 0 = 176\) feet per second.
02

Determine the Exponential Formula for D

The velocity after 2 seconds \(v(2)\) is 54.75 feet per second, which implies the difference \(D(2) = T - v(2) = 176 - 54.75 = 121.25\) feet per second. Since \(D\) is an exponential function of time \(t\), it can be represented as \(D = D_0 e^{-kt}\), where \(D_0\) is the initial value \(176\). We have \(D(2) = 176 e^{-2k} = 121.25\). Solving for \(k\) gives us \(e^{-2k} = \frac{121.25}{176}\). Taking the natural logarithm on both sides, \(-2k = \ln \left( \frac{121.25}{176} \right)\), hence \(k = -\frac{1}{2} \ln \left( \frac{121.25}{176} \right)\). Substituting back, \(D(t) = 176 e^{-kt}\) gives the exponential formula for \(D\).
03

Express Formula for Velocity v

Since \(D = T - v\), we can write \(v = T - D\). Plugging the formula for \(D\) from Step 2, we have \(v(t) = 176 - 176 e^{-kt}\).
04

Calculate Velocity at 4 Seconds

Substitute \(t = 4\) into the velocity formula from Step 3: \(v(4) = 176 - 176 e^{-4k}\). Since \(k\) was calculated before, substitute \(k\) to find \(v(4)\). This involves evaluating the expression \(176 - 176 e^{-4k}\) using the value of \(k\) determined in Step 2.

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Key Concepts

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

Terminal Velocity
When a skydiver jumps from an airplane, there's a point during the fall where the forces acting on them balance out, causing their velocity to stop increasing. This constant speed is known as "terminal velocity." It happens because the force of gravity pulling the skydiver downwards is exactly balanced by the air resistance pushing upwards.

Terminal velocity depends on several factors:
  • The body's shape and size 鈥� larger surface area experiences more air resistance.
  • The object's mass 鈥� heavier objects may have a higher terminal velocity.
  • The density of the fluid the body moves through 鈥� affects the amount of drag.
For an average human, terminal velocity is approximately 176 feet per second when falling through the air. At this speed, the skydiver falls steadily without speeding up.
Velocity
Velocity is the speed of something in a given direction. It's an important concept when discussing motions such as skydiving.

When you jump out of an airplane, your initial velocity is zero since you start from rest. As you fall, your velocity increases due to gravity. It increases until it reaches terminal velocity, where it becomes constant as the upward force of air drag cancels out gravitational pull.

In mathematical terms, you can express velocity (v) using the terminal velocity (T) and the difference (D):
  •  v(t) = T - D(t) 
Here,  t  is time, and the formula also considers how D changes exponentially over time.
Exponential Growth and Decay
Exponential functions describe processes that grow or shrink at rates proportional to their size. In the context of a falling skydiver, the difference (D) between terminal velocity and actual velocity decreases exponentially over time as the skydiver approaches terminal velocity.

The formula that governs this exponential decay is:
  • D(t) = D_0 e^{-kt}
Where D_0 is the initial difference, k is the decay constant, and t is time.

This mean, over time, D(t) becomes smaller, ultimately approaching zero as the skydiver nears terminal velocity. The decay constant k dictates how quickly this happens, based on factors like gravity and air resistance.
Mathematical Modeling
Mathematical modeling involves using equations to represent real-world scenarios. This powerful tool helps us understand phenomena such as skydiving without physically recreating the conditions.

For a skydiver, we describe their velocity and how it changes using exponential functions. The difference in velocity D and its dependence on time illustrates how exponential decay helps model the skydiver's motion.
Mathematical models offer:
  • Predictive capabilities 鈥� forecast results under various conditions.
  • Visualization 鈥� see how variables interact over time.
  • Problem-solving 鈥� assist in finding optimal solutions to real-life problems.
Using these models, we can provide accurate descriptions of complex behaviors like that of a skydiving journey. By understanding these models, you gain insights and solutions far beyond simple predictions.

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Most popular questions from this chapter

The rate of growth \(G\), in thousands of dollars per year, in sales of a certain product is a function of the current sales level \(s\), in thousands of dollars, and the model uses a quadratic function: $$ G=1.2 s-0.3 s^{2} . $$ The model is valid up to a sales level of 4 thousand dollars. a. Draw a graph of \(G\) versus \(s\). b. Express using functional notation the rate of growth in sales at a sales level of \(\$ 2260\), and then estimate that value. c. At what sales level is the rate of growth in sales maximized?

By 1619 Johannes Kepler had completed the first accurate mathematical model describing the motion of planets around the sun. His model consisted of three laws that, for the first time in history, made possible the accurate prediction of future locations of planets. Kepler's third law related the period (the length of time required for a planet to complete a single trip around the sun) to the mean distance \(D\) from the planet to the sun. In particular, he stated that the period \(P\) is proportional to \(D^{1.5}\). a. Neptune is about 30 times as far from the sun as is the Earth. How long does it take Neptune to complete an orbit around the sun? (Hint: The period for the Earth is 1 year. If the distance is increased by a factor of 30 , by what factor will the period be increased?) b. The period of Mercury is about 88 days. The Earth is about 93 million miles from the sun. How far is Mercury from the sun? (Hint: The period of Mercury is different from that of the Earth by a factor of \(\frac{88}{365}\).)

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