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Chapter 11: Q. 47 (page 872)

Use the given velocity vectors v(t)=r'(t)and initial positions in exercise to find the position function r(t).

localid="1650736930792" v(t)=eti+lntj,r(1)=i-6j

Short Answer

Expert verified

r(t)=(et+1-e)i+(tlnt-t-5)j

Step by step solution

01

Given Information

Consider v(t)=eti+lntj,r(1)=i-6j

The objective is to find the position function r(t).

since v(t)=r'(t)

we have

r(t)=鈭�v(t)dt=鈭�(eti+lntj)dt=i鈭�etdt+j鈭�lntdt=eti+(tlnt-t)j+c

Where c is a vector constant.

=(et+c1)i+(tlnt-t+c2)jwherec1andc2are scalars.

02

Expression

So

r(t)=(et+c1)i+(tlnt-t+c2)jr(1)=(e1+c1)i+(1ln1-1+c2)jr(1)=(e+c1)i+(-1+c2)j...(1)

but the given initial position r(t)=i-6j.........(2)

Comparing (1) and (2)

e+c1=1,-1+c2=-6c1=1-e,c2=-6+1c1=1-e,c2=-5r(t)=(et+1-e)i+(tlnt-t-5)j

03

Calculation

Check: Taking the derivative of r(t)=(et+1-e)i+(tlnt-t-5)j,

We obtain r'(t)=eti+lntj, which is the correct tangent vector function. Next, we evaluate

r(1)=(et+1-e)t+(1ln1-1-5)j=j-6j

Which is true according to the problem.

Thusr(t)=(et+1-e)i+(tlnt-t-5)j

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Most popular questions from this chapter

Let r(t)=x(t),y(t),z(t)be a differentiable vector function on some interval I鈯�Rsuch that the derivative of the unit tangent vector T(t0)=0, where t0鈭�I. Prove that the binormal vectorB(t0)=T(t0)脳N(t0)

(a) B(t0)is a unit vector;

(b)B(t0)is orthogonal to both T(t0)and N(t0).

Also, prove that T(t0),N(t0), and B(t0)form a right-handed coordinate system.

Prove that the tangent vector is always orthogonal to the position vector for the vector-valued function.

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x鈥�(t)=x,y鈥�(t)=x2,x(0)=1,y(0)=2. (HINT: Start by solving the initial-value problem x鈥�(t)=x,x(0)=1.)

Completethefollowingdefinition:Ifr(t)=x(t),y(t),z(t)isatwice-differentiablepositionfunction,thentheaccelerationvectora(t)is....

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