Q. 35 Let r(t)聽be a vector function w... [FREE SOLUTION]

Chapter 11: Q. 35 (page 889)

Let $r (t)$ be a vector function whose graph is a space curve containing distinct pointsP and Q. Prove that if the acceleration is always 0, then the graph of r is a straight line.

Short Answer

Expert verified

If $a (t) = <0, 0, 0>$ , then the velocity vector is $v (t) = <伪, 尾, 纬>$ for constants $伪, 尾, 纬$ . The position function will be $r (t) = (c_{1} t + 伪) + (c_{2} t + 尾) + (c_{3} t + 纬)$ , where $c_{1}, c_{2}, c_{3}$ are three more constants. The graph of $r (t)$ is a straight line.

Step by step solution

Step 1. Given information.

Consider the given question,

$r (t)$ is a vector function whose graph is a space curve containing distinct pointsP and Q.

Step 2. Assume the acceleration vector to be at=0,0,0.

Assume acceleration vector, $a (t) = <0, 0, 0>$ . Then,

localid="1649614922270" $v (t) = 鈭� a (t) d t v (t) = 鈭� <0, 0, 0> d t v (t) = 鈭� 0 d t i + 鈭� 0 d t j + 鈭� 0 d t k$

Integral of a vector function,

$v (t) = c_{1} i + c_{2} j + c_{3} k v (t) = <c_{1}, c_{2}, c_{3}>$

Step 3. Consider rt=∫vt dt.

Consider $r (t) = 鈭� v (t) d t$ ,

localid="1649615120747" $r (t) = 鈭� <c_{1}, c_{2}, c_{3}> d t r (t) = 鈭� c_{1} d t i + 鈭� c_{2} d t j + 鈭� c_{3} d t k r (t) = (c_{1} t + 伪) + (c_{2} t + 尾) + (c_{3} t + 纬) k$

localid="1649615158797" $r (t) = (c_{1} t + 伪) + (c_{2} t + 尾) + (c_{3} t + 纬)$ is in linear form.

Hence, $r (t)$ represents a straight line.

Thus, if the acceleration of $r (t)$ is $0$ , then the graph ofr is a straight line.

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91影视

Let $r (t)$ be a vector function whose graph is a space curve containing distinct pointsP and Q. Prove that if the acceleration is always 0, then the graph of r is a straight line.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Assume the acceleration vector to be at=0,0,0.

Step 3. Consider rt=∫vt dt.

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91影视

Let r(t)be a vector function whose graph is a space curve containing distinct pointsP and Q. Prove that if the acceleration is always 0, then the graph of r is a straight line.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Assume the acceleration vector to be at=0,0,0.

Step 3. Consider rt=∫vt dt.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let $r (t)$ be a vector function whose graph is a space curve containing distinct pointsP and Q. Prove that if the acceleration is always 0, then the graph of r is a straight line.