Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.96c
Chapter 7, Problem 7.3.96c

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

Verified step by step guidance
1
Recall that the velocity function \(v(t)\) is the derivative of the position function \(s(t)\) with respect to time \(t\). So, start by writing down the velocity function \(v(t) = s'(t)\).
From Exercise 95, the position function for a falling body with mass \(m\) and drag coefficient \(k\) is typically given by \(s(t) = \frac{mg}{k}t - \frac{m^2g}{k^2}(1 - e^{-\frac{k}{m}t})\). Differentiate this function to find the velocity function \(v(t)\).
After differentiating, you should get \(v(t) = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\). This formula shows how velocity changes over time considering air resistance.
Set the velocity equal to 45 m/s to find the time \(t\) when the BASE jumper reaches this speed: \(45 = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\).
Solve the equation for \(t\) by isolating the exponential term, then taking the natural logarithm: first, rearrange to get \(e^{-\frac{k}{m}t} = 1 - \frac{45k}{mg}\), then take \(\ln\) on both sides and solve for \(t\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position and Velocity Functions

The position function describes the location of a falling body over time, while the velocity function is its derivative, representing the rate of change of position. Understanding how to differentiate the position function is essential to find the velocity at any given time.
Recommended video:
10:17
Using The Velocity Function

Solving for Time Given Velocity

To find the time when the velocity reaches a specific value, set the velocity function equal to that value and solve for time. This often involves algebraic manipulation or solving transcendental equations depending on the form of the velocity function.
Recommended video:
10:17
Using The Velocity Function

Parameters in Motion Equations (Mass and Drag Coefficient)

Mass (m) and drag coefficient (k) affect the motion of the falling body by influencing acceleration and terminal velocity. These parameters appear in the differential equations governing motion and must be used correctly to model realistic velocity and position.
Recommended video:
Guided course
04:11
Eliminate Parameter: Equations with Trig
Related Practice
Textbook Question

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


c. (exp(x))ᵖ = exp(px), p rational

62
views
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln(1 + √2) = −ln(−1 + √2)

53
views
Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

43
views
Textbook Question

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

c. csch⁻¹ 5

59
views
Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

35
views
Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

100
views