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Chapter 11: Q. 2TF (page 891)

A decomposition of the acceleration vector: Find compv(t)a(t), where v anda are the velocity and acceleration vectors, respectively, of the following functions.

r(t)=cost,sint,t

Short Answer

Expert verified

The component of a(t)alongv(t)iscompv(t)a(t)=sin2t.

Step by step solution

01

Step 1. Given Information.

The given function isr(t)=cost,sint,t.

02

Step 2. Calculating the velocity and acceleration vector. 

Let's calculate the given function,

r(t)=cost,sint,tr'(t)=v(t)=-sint,cost,1r''(t)=a(t)=-cost,-sint,0Now,v(t)=-sint2+cost2+12=sin2t+cos2t+1=2........(a)Let'sfindat路vt=-cost,-sint,0路-sint,cost,1=2sintcost..........(b)

03

Step 3. Solve.

Put (a) and (b) in compv(t)a(t)=a(t).v(t)v(t),

compv(t)a(t)=2sintcost2compv(t)a(t)=2sintcost2compv(t)a(t)=2sintcostcompv(t)a(t)=sin2t

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Most popular questions from this chapter

Let r(t)=x(t),y(t),z(t),t鈭�[a,鈭�), be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in some sphere centered at the origin. (Hint: Consider the functions r1(t)=cost,sint,1/tand r2(t)=cost,sint,t,bothwithdomain[1,鈭�).)

For each of the vector-valued functions, find the unit tangent vector.

r(t)=(t,t2,t3)

Show that the graph of the vector function r(t)=鉄�3sin鈦�t,5cos鈦�t,4sin鈦�t鉄�is a circle. (Hint: Show that the graph lies on a sphere and in a plane.)

Let r(t)=x(t),y(t),t鈭�[a,鈭�), 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 鈫� 鈭�? Provide an example illustrating your answer.

Let t=f(蟿)be a differentiable real-valued function of 蟿, and let r(t)be a differentiable vector function with three components such that f(蟿)is in the domain of rfor every value of 蟿on some interval I. Prove that drd蟿=drdtdtd蟿. (This is Theorem 11.8.)
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