Position, velocity, and the length of a path

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

For r(t) = (3t, 4t, 0), what is the speed?
Answer: ______________
Sam says the velocity comes from differentiating each component of position. Is Sam right?
Circle one: True False
For r(t) = (3t, 4t, 0), what is the speed?
Answer: ______________
If r(t) = (t, t squared), what is its velocity vector?
- (1, 2t)
- (1, t squared)
- (2t, 1)
For r(t) = (2t, 3t, 6t), what is the speed?
Answer: ______________
For r(t) = (3t, 4t, 0) with 0 <= t <= 2, what is the arc length?
Answer: ______________
For r(t) = (3t, 4t, 0), what is the unit tangent vector?
- (3/5, 4/5, 0)
- (3, 4, 0)
- (1, 0, 0)
- (3/5, 4/5, 1)
For r(t) = (2t, 3t, 6t), what is the unit tangent vector?
- (2, 3, 6)
- (1/3, 1/3, 1/3)
- (2/7, 3/7, 6/7)
For r(t) = (2t, 3t, 6t) with 0 <= t <= 3, what is the arc length?
Answer: ______________
A particle moves with constant speed 5 from t = 0 to t = 2 along a curving path. Ana says the arc length is still 10. Is Ana right?
Circle one: True False

Answer key

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

5 · Velocity is (3, 4, 0), and its length is 5.
True · Sam is right: velocity is the componentwise derivative of position.
5 · Velocity is (3, 4, 0) with length 5.
(1, 2t) · Differentiate each piece: t gives 1 and t squared gives 2t.
7 · Velocity (2, 3, 6) has length sqrt(4 + 9 + 36) = 7.
10 · Constant speed 5 over 2 time units gives length 10.
(3/5, 4/5, 0) · Divide velocity (3, 4, 0) by speed 5.
(2/7, 3/7, 6/7) · Velocity (2, 3, 6) has length 7, so divide each component by 7.
21 · Speed 7 is constant, so length is 7 times 3.
True · Length only cares about speed, not the shape of the path. 5 times 2 is 10.