# SOLUTION: ENGM 2101 Dalhousie Universit yApplied Vector Calculus Quiz Questions

Dalhousie University
Department of Engineering Mathematics
ENGM 2101 Applied Vector Calculus Summer 2020
Quiz #1
Faculty of Engineering Academic Uprightness Proposition for Inclusion in Exams
This is a explicit proposition of commitment to academic uprightness for impost in ENGM 2101
Applied Vector Calculus Summer 2020. Students are asked to corroborate their commitment to the
principles of ethics, uprightness and professionalism embedded in this proposition as it is an
expectancy of all novices and homogeneous weighty, an expectancy of you during your coming walk
as a Professional Engineer.
Expectation of Student:
The exam is to be accomplishedd partially by the novice. Specifically, novices are not to discuss
the exam behind a while others or cooperate on the substance of the exam behind a while others, including other
students in this series, in any habit. Students are expected to flourish the instructions on the
exam and use singly the series symbolical or instrument as exempt. A commitment to a high
standard of separate uprightness and professionalism is expected of all engineers and is homogeneous
expected of those inaugurated towards a walk in engineering.
Certification
By yieldting this product, I avow that the product represents singly my own efforts and was
completed in a habit agreeing behind a while the instructions supposing. I corroborate that I comprehend the
meaning and consequences of imposture, plagiarism and other academic offences beneath the
Faculty Discipthread Procedures for Academic Dishonesty , and am conscious of my responsibilities
beneath the Intellectual Honesty Policy .
1
Instructions Specific to This Quiz
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This satire is beneficial from 11am Tuesday May 12th until 11am Wednesday May 13th.
The satire is contrived to be accomplishedd behind a whilein 2 hrs, but you may exhaust further or hither time
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While not required to accomplished the satire, you are agreeable to advert to your notes, the
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Attempt all 5 questions
There are a aggregate of 55 subject-matters
The subject-matter appraise of each question/sub-question is ardent in parentheses
If you get store, establish an certainty, propound it, and proceed
Partial reputation may be ardent for inexact exculpations that prove punish method
Show your product – no reputation procure be ardent for exculpations behind a whileout adapted justification
2
Question 1 (1 subject-matter each = 10 subject-matters aggregate)
State whether the flourishing are frequently penny (T), frequently sham (F), or there is not enough
counsel to establish the vill (N).
(
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a) r1 (t) = ( cos(3t),sin(3t) ) and r2 (t) = cos(t 2 ),sin(t 2 ) portray the identical flexion for t ≥ 0 .
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b) The tangent thread for r (t) at the subject-matter t = t0 is ardent by (x, y, z) = r (t0 ) + k r ′′(t0 ) , where k is a
scalar.
⌢
dTˆ
dTˆ
c) Ardent T (s) , where s = f (t) , then
and
are correlative.
ds
dt
d) The part regular N̂ is orthogonal to
dN̂
.
dt
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e) The concatenate of soaring for a missile that is instituted from the reason behind a while a quickness v is longer
than the concatenate of soaring of a irrelative missile that is instituted from a culmination h over the reason
behind a while the identical quickness.
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f) The interval concurrently r (t) betwixt a ≤ t ≤ b is shorter than the majority of displacement
!
!
betwixt r (a) and r (b) .
g) There is a element of aid that is correlative to the part binormal.
h) The tangential element of aid represents the rate of diversify of address.
i) The utmost inflexion of a missile’s trajectory occurs at the utmost culmination of the
projectile.
j) The torsion of a helix is cipher.
Question 2 (9 subject-matters)
a – 2 subject-matters) Ardent that a atom starts at P0 = (2, 1, 1), voyages in a right thread and gets to P1 =
(-1, 0, 4) behind 1 cooperate, and it continues concurrently the identical thread at the identical hurry, where procure the
atom be 2 cooperates behind it passed P1?
Question 2 continued on the next page
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b – 2 subject-matters) Select the discretion that portrays the quantity flexion shown beneath
(
)
(
)
i) ( acost,asint,bt ) , ii) e at require,e at sint,bt , iii) ( bt,acost,asint ) , iv) bt,e at require,e at sint , or
v) none of the over
⎛ t 2 −t 3 ⎞
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c) Two atoms voyage concurrently flexions r1 (t) = ⎜ t, ,
and r1 (t) = 22 − 3t,2 + t,4 − 2t both
⎟
⎝ 2 8 ⎠
beginning at t = 0. The trajectories iteme at the subject-matter (4, 8, -8), but the atoms do not
collide.
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i -- 2 subject-matters) Ascertain the equation of the tangent thread for r1 at the subject-matter of itemeion.
ii -- 3 subject-matters) Ascertain the inclination at which the two flexions iteme.
Question 3 (6 subject-matters)
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a) Ardent a atom is on the trajectory r (t)= ( sint,cost,sin(2t) ) for 0 ≤ t ≤ 2π ,
i -- 2 subject-matters) Ascertain an countenance for the hurry as a employment of time
ii -- 2 subject-matters) Determine the utmost hurry of the atom.
d !
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r (t) × v(t) ) = r (t) × a(t) , where v(t) is the quickness, and a(t) is
(
dt
the aid, and excuse your exculpation.
b -- 2 subject-matters) Propound whether
4
Question 4 (11 subject-matters)
(
)
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a -- 3 subject-matters) Ardent an aid a = 1,e2t ,4t and judicious quickness v(0) = ( 2,1,0 ) , ascertain the
displacement betwixt t = 0 and t = 1.
b -- 4 subject-matters) A wildlife conservationist wants to inspirer a tranquilizer fly at an endangered monkey
sitting at the top of a 20m tree, which is 80 m abroad. She knows the monkey procure faint from the
tree the force the gun is inspirerd, and boon her gun accordingly. Ardent that the stop quickness is
100m/s, ascertain the culmination at which the fly hits the monkey.
(
)
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c -- 4 subject-matters) Ardent a atom voyages concurrently the flexion r (t) = t 2 , 2t, lnt , ascertain the interval
traveled for 1 ≤ t ≤ 2 .
Question 5 (19 subject-matters)
a -- 3 subject-matters) For the flexion shown beneath, outline the part tangent, Tˆ , and part regular, N̂ ,
vectors at the two subject-matters implied (where t1
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