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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.96c
Capitolo 7, Problema 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)?

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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 \(t\).

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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.
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Percorso guidato
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.
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Percorso guidato
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.
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04:11
Eliminate Parameter: Equations with Trig
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