Arc Length and Curvature of a Space Curve · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How sharply a space curve bends

Mathematics · Geometry · ages 20-21
Name ______________________   Date ____________

  1. A particle follows r(t) = (t, 2t, 2t) for t from 0 to 2. How far does it travel?

    Answer: ______________

  2. A circle has radius 4. What is its curvature?

    Answer: ______________

  3. A circle has radius 4. What is its curvature?

    Answer: ______________

  4. A particle follows r of t equal (t, 2t, 2t) for t from 0 to 2. How far does it travel?

    Answer: ______________

  5. A car rounds a bend at steady speed with its acceleration pointing toward the inside of the bend.

    Circle one:   True   False

  6. A rocket speeds up along a straight track. Which component of its acceleration is nonzero?

    • The tangential component
    • The normal component
    • Both components
    • Neither component

  7. A car rounds a bend at steady speed. Zoe says the acceleration points toward the inside of the bend. Is Zoe right?

    Circle one:   True   False

  8. A circle has radius 5. What is its curvature? Give a decimal.

    Answer: ______________

  9. A circle has radius 10. What is its curvature? Give a decimal.

    Answer: ______________

  10. Two drivers trace the same bend, one faster. How do the curvatures compare?

    • Faster means larger curvature
    • Slower means larger curvature
    • Equal, since curvature ignores tracing speed
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Answer key

For grown-ups. Fold this page away before handing over the rest.

How sharply a space curve bends W1-mt_8mXYE8bVG1-s1

  1. 6 · The velocity is (1, 2, 2) with speed 3, and 3 times the parameter interval of length 2 gives 6.
  2. 0.25 · For a circle, curvature is 1 over the radius, so 1/4 = 0.25. Tighter circles bend more.
  3. 0.25 · For a circle, curvature is 1 over the radius.
  4. 6 · Velocity (1, 2, 2) has speed 3 over an interval of length 2.
  5. True · Steady speed kills the tangential part, leaving the inward normal part.
  6. The tangential component · Speeding up on a straight track changes speed but not direction, so only the tangential component is nonzero.
  7. True · Zoe is right. At steady speed the tangential part is zero, and the remaining normal part points to the inside of the bend.
  8. 0.2 · Curvature is 1 over the radius, so 1 over 5.
  9. 0.1 · 1 over 10 is 0.1; wider circles bend less.
  10. Equal, since curvature ignores tracing speed · Curvature is a property of the path, not of the speed along it.
Worksheet · LightMySky