In this problem, answer \"True\" or \"False\" for each question. Note: there is

Question

In this problem, answer "True" or "False" for each question.

Note: there is no partial credit for this problem. You must answer all parts correctly to receive credit. You will not be shown the correct answers for individual parts.

Acceleration in space has tangential, normal and binormal components.
True
False

Consider a space curve defined by a smooth vector function r(t). If r(t)=0 for all t, then the curve is a straight line.
True
False

Consider a space curve defined by a smooth vector function r(t). If the curve is a straight line, then r(t)=0 for all t.
True
False

If a smooth space curve has constant curvature, then it is a circle.
True
False

Suppose r (t)=cos(t)i+sin(t)j+5tk represents the position of a particle on a helix, where z is the height of the particle.

(a) What is t when the particle has height 15?
  

(b) What is the velocity of the particle when its height is 15?


(c) When the particle has height 15, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line.

Explanation / Answer

(1) Acceleration in space has tangential, normal and binormal components.

We know that acceleration in binormal direction is always zero. Therefore, Acceleration has only two components, namely, tangential and normal. Therefore, this statement is false.

(2) Consider a space curve defined by a smooth vector function r(t). If r(t)=0 for all t, then the curve is a straight line.

We know that r''(t) will only be zero (in 2 D space) if both x and y components in r'(t) are either constant or zero or a combination of these. If r'(t) has zero components, then r(t) will not be a staight line. Therefore this statement is not always true. Hence it is false.

(3) Consider a space curve defined by a smooth vector function r(t). If the curve is a straight line, then r(t)=0 for all t.

Since r(t) is the curve of a straight line, therefore, we can write r(t) = <t, mt+c>

r'(t) = <1, m>

r''(t) = <0, 0 >

Therefore, r''(t) = 0

Therefore, the given statement is true.

(4) If a smooth space curve has constant curvature, then it is a circle.

A smooth space ruve with constant curvature is a line. Therefore this statement is false.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.