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Chapter 11: Q. 71 (page 873)

Prove that if a particle moves along a curve at a constant speed, then the velocity and acceleration vectors are orthogonal.

Short Answer

Expert verified

Ans:

ddtr'2=ddtr'·r'=0

Step by step solution

01

Step 1. GIven information: 

Consider a twice differentiable vector function r(t).let a particle moves along the curve at a constant speed.

02

Step 2. Theory:

Here we assume that the velocity and acceleration vectors are orthogonal and prove that the particles move along r(t)at a constant speed. That is, we show that r'(t)is constant. Since the velocity and acceleration vectors are orthogonal, their dot product is zero. That is we have

v(t)·a(t)=0⇒r'(t)·r''(t)=0…….(1)

03

Step 3. Proving: 

Consider

ddtr'2

=ddtr'·r'

=r'ddtr'+r'ddtr'Derince ‖r‖2=r·r

=r'·r''+r'·r''ddtr'=r''

=2r'-r'' add from equations(1) product of 2 function

=2(0)multiply

=0

ddtr'2=0.that means r' is a constant. That is, the magnitude of the velocity vector is

constant.

Thus the particle movesalong r(t) at a constant speed.

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Most popular questions from this chapter

Let r1(f)and r2(t) be differentiable vector functions with three components each. Prove that

ddtr1(t)-r2(t)=r1'(t)-r2(t)+r1(t)-r2'(t)∣(This is Theorem 11.11 (c).)

Evaluate and simplify the indicated quantities in Exercises 35–41.

(1,t,t2)×(1,t2,t3)

Given a differentiable vector-valued function r(t), what is the relationship between r't0and Tt0at a pointt0in the domain of r(t)?

Let r(t)=x(t),y(t),t∈[a,∞), be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t → ∞? Provide an example illustrating your answer.

Let y = f (x). State the definition for the continuity of the function f at a point c in the domain of f .
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