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Exercise 3. Read the boxed definitions on p. 767. According to these definitions, how are the  concepts of velocity and speed defined here different?

Exercise 3. Read the boxed definitions on p. 767. According to these definitions, how are the  concepts of velocity and speed defined here different?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 3. Read the boxed definitions on p. 767. According to these definitions, how are the 
concepts of velocity and speed defined here different?

N FIGURE 13.7 The curve and the velocity vector when - 77/4 for the motion given in Example 4. Solution The velocity and acceleration vectors at time i are and the speed is v(t) = r' (t) = -2 sin ti + 2 cos tj 10 cos t sin t k = -2 sin ti + 2 cos tj - 5 sin 2rk, a(t) = r"(t) = -2 costi - 2 sin tj- 10 cos 21 k. |v(t) = √(-2 sin t)² + (2 cos t)² + (−5 sin 2r)² = V4 + 25 sin² 2t. When 1 = 77/4, we have 7T 4 √29. A sketch of the curve of motion, and the velocity vector when t = 77/4, can be seen in Figure 13.7. √21 + √2j + 5k, 7T 4 = -√2i + √2j. 7 T |(¹)|- We can express the velocity of a moving particle as the product of its speed and direction: Velocity = = (speed) (direction). |v|| Differentiation Rules Because the derivatives of vector functions may be computed component by component, the rules for differentiating vector functions have the same form as the rules for differenti- ating scalar functions.
Transcribed Image Text:N FIGURE 13.7 The curve and the velocity vector when - 77/4 for the motion given in Example 4. Solution The velocity and acceleration vectors at time i are and the speed is v(t) = r' (t) = -2 sin ti + 2 cos tj 10 cos t sin t k = -2 sin ti + 2 cos tj - 5 sin 2rk, a(t) = r"(t) = -2 costi - 2 sin tj- 10 cos 21 k. |v(t) = √(-2 sin t)² + (2 cos t)² + (−5 sin 2r)² = V4 + 25 sin² 2t. When 1 = 77/4, we have 7T 4 √29. A sketch of the curve of motion, and the velocity vector when t = 77/4, can be seen in Figure 13.7. √21 + √2j + 5k, 7T 4 = -√2i + √2j. 7 T |(¹)|- We can express the velocity of a moving particle as the product of its speed and direction: Velocity = = (speed) (direction). |v|| Differentiation Rules Because the derivatives of vector functions may be computed component by component, the rules for differentiating vector functions have the same form as the rules for differenti- ating scalar functions.
Chapter 13 Vector-Valued Functions and Motion in Space 13.1 Curves in Space and Their Tangents DEFINITIONS If r is the position vector of a particle moving along a smooth curve in space, then v(1) dt is the particle's velocity vector, tangent to the curve. At any time 1, the direction of v is the direction of motion, the magnitude of v is the particle's speed, and the derivative a = dv/dt, when it exists, is the particle's acceleration vector. In summary, 1. Velocity is the derivative of position: 2. Speed is the magnitude of velocity: 3. Acceleration is the derivative of velocity: Speed v = 767 dvd²r dt² dt 4. The unit vector v/v is the direction of motion at time t. EXAMPLE 4 Find the velocity, speed, and acceleration of a particle whose motion in space is given by the position vector r(t) = 2 costi + 2 sin tj + 5 costk. Sketch the velocity vector v(7#/4).
Transcribed Image Text:Chapter 13 Vector-Valued Functions and Motion in Space 13.1 Curves in Space and Their Tangents DEFINITIONS If r is the position vector of a particle moving along a smooth curve in space, then v(1) dt is the particle's velocity vector, tangent to the curve. At any time 1, the direction of v is the direction of motion, the magnitude of v is the particle's speed, and the derivative a = dv/dt, when it exists, is the particle's acceleration vector. In summary, 1. Velocity is the derivative of position: 2. Speed is the magnitude of velocity: 3. Acceleration is the derivative of velocity: Speed v = 767 dvd²r dt² dt 4. The unit vector v/v is the direction of motion at time t. EXAMPLE 4 Find the velocity, speed, and acceleration of a particle whose motion in space is given by the position vector r(t) = 2 costi + 2 sin tj + 5 costk. Sketch the velocity vector v(7#/4).
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