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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 68b
Capitolo 4, Problema 68b

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .


b. When do the particles collide?

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To find when the particles collide, we need to determine when their positions are equal. Set the position functions equal to each other: \( \cos t = \cos (t + \frac{\pi}{4}) \).
Recall the trigonometric identity for cosine: \( \cos A = \cos B \) implies \( A = 2n\pi \pm B \) for integer \( n \). Apply this identity to the equation \( \cos t = \cos (t + \frac{\pi}{4}) \).
This gives us two possible equations: \( t = 2n\pi + (t + \frac{\pi}{4}) \) and \( t = 2n\pi - (t + \frac{\pi}{4}) \).
Solve each equation for \( t \). For the first equation, simplify to find \( 0 = 2n\pi + \frac{\pi}{4} \), which is not possible. For the second equation, simplify to find \( 2t = 2n\pi - \frac{\pi}{4} \).
Solve the second equation for \( t \) to find the specific times when the particles collide. This will give you the values of \( t \) in terms of \( n \).

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Particle Position Functions

In this context, the positions of the two particles are given by the functions s₁ = cos(t) and s₂ = cos(t + π/4). These functions describe how the position of each particle changes over time, with 't' representing time. Understanding these functions is crucial for determining when the particles are at the same position, which is essential for solving the collision problem.
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Relations and Functions

Collision Condition

For two particles to collide, their positions must be equal at the same time. This means we need to set the position functions equal to each other: cos(t) = cos(t + π/4). Solving this equation will yield the values of 't' at which the particles occupy the same position on the s-axis, indicating a collision.
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Initial Value Problems Example 1

Trigonometric Identities

Trigonometric identities, such as the cosine addition formula, are essential for simplifying and solving equations involving trigonometric functions. In this case, using the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b) can help in transforming the collision condition into a more manageable form, allowing for the determination of the specific times 't' when the particles collide.
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Verifying Trig Equations as Identities
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