Deck 12: Vector Functions and Curves

Question 1

Let r(t) = 4t i + 3sin(t) j - 3cos(t) k be a differentiable vector function giving the position r of a particle at time t. Find the speed of the particle at time t = seconds.

A) 5.5 units/s
B) 5.0 units/s
C) 4.5 units/s
D) 4.0 units/s
E) 3.5 units/s

Answer 1

5.0 units/s

Question 2

Find the velocity at time t = -2 of a particle whose position at time t is given by r(t) = 3t i - j.

A) v(-2) = - 3i + 12j
B) v(-2) =- 3i - 12j
C) v(-2) =3i + 12j
D) v(-2) = 3i - 12j
E) v(-2) = 3i - 4j

Answer 2

v(-2) = 3i - 12j

Question 3

Find the acceleration at time t = -1 of a particle whose position at time t is given by r(t) = 4t i + j.

A) a(-1) = -2j
B) a(-1) = 2j
C) a(-1) = i - 2j
D) a(-1) = i + 2j
E) a(-1) = 2i + j

Answer 3

a(-1) = 2j

Question 4

A moving particle starts at an initial position (1, 0, 0) with initial velocity i - j + k. Its acceleration at time t is a(t) = 4t i + 6t j + k. Find its velocity and position at time t > 0.

A)
B)
C)
D)
E)

Question 5

Find the velocity, speed, and acceleration at time t of a particle that has position function r(t) = (2sin t) i + 6t j + (2cos t) k.

A)
B)
C)
D)
E)

Question 6

Find the velocity, speed, and acceleration at time t of a particle that has position function r(t) = t i + j + k.

A)
B)
C)
D)
E)

Question 7

What kind of curve is r(t) = (sin t) i + (sin t) j + 2(cos t) k?

A) a circular helix
B) an oval plane curve that is, however, not an ellipse
C) an ellipse (but not a circle)
D) a circle
E) a parabola

Question 8

Describe the curve r(t) = (sin t) i + (cos t) j + k.

A) a helix wound around the cylinder + = 1
B) the circle (of radius 1) in which the plane z = intersects the sphere + + = 4
C) the circle (of radius ) in which the plane z = 1 intersects the sphere + + = 3
D) the circle (of radius ) in which the plane z = intersects the cylinder + = 1
E) a helix wound around the cylinder + = 3

Question 9

Find an equation of the line tangent to the parametric space curve
r(t) = ( + 3t + 1) i + (2 - 7t j + (4sin(t) -3)k at the point on the curve corresponding to t = 0.

A) r(u) = (1 + 3u) i + (2 - 7u) j + (- 3 + 4s) k, u R
B) 3x - 7y + 4z -23 = 0
C) x + y + z = 0
D) r(u) = (3 + u) i + (-7 + 2u) j + (4 - 3u) k, u 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 R
E) x + 2y - 3z - 14 = 0

Question 10

A particle is moving to the right with constant speed 2 along the curve y = cos x in the xy-plane. Find its velocity at the instant when it crosses the vertical line x = .

A) v = i - j
B) v = i - j
C) v = i - j
D) v = i - j
E) v = i + j

Question 11

The position vector of a moving particle in space is given by the vector equation . When will the speed of the particle be 4 units?

A) t = 6 and t = -2
B) t = 0
C) t = -6 and t = 2
D) t = 6
E) t = 2

Question 12

An object is moving to the right in the xy-plane along the curve y = . At the instant when it is at the point , its speed is 2 and that speed is decreasing at rate 1. What are the velocity and acceleration of the object at that instant?

A) v = (i + j), a = (1 + ) i + (1 - ) j
B) v = (i + j), a = i - j
C) v = 2 (i + j), a = (1 + ) i + (1 - ) j
D) v = 2 (i + j), a = -i + j
E) v = (i + j), a = -i + j

Question 13

At a certain instant a moving object has velocity v = 2i - 3j - k (in cm/s) and acceleration . How fast is its speed changing at that instant?

A) speed is decreasing at / cm
B) speed is increasing at / cm
C) speed is decreasing at / cm
D) speed is increasing at / cm
E) speed is unchanging

Question 14

Suppose that the position r(t) and velocity v(t) of a moving object satisfy r(t) . v(t) = 0 for all t. What does this imply about the curve r(t)?

A) The curve lies on a sphere centred at the origin.
B) The curve lies in a plane.
C) The curve is a circle.
D) The curve is a helix.
E) The curve lies on a hyperboloid.

Question 15

The position function of a particle is given by r(t) = , When is the speed a minimum?

A) t = 2
B) t = 3
C) t = 4
D) t = 5
E) t = 1

Question 16

The position of a particle at time t is given by r = (cos t) i + (sin t) j + t k. Find the speed and the magnitude of the acceleration at any time t. Describe the motion.

A) = ; = ; path is a circular helix
B) = ; = ; path is a circle
C) = ; = 2; path is a circular helix
D) = ; = 1; path is a circular helix
E) = ; = 1; path is a circle

Question 17

Solve the initial-value problem

A) r(t) = cos(kt) - sin(kt)
B) r(t) = cos(kt) + sin(kt)
C) r(t) = cos(kt) + sin(kt)
D) r(t) = cos(kt) - sin(kt)
E) r(t) = sin(kt) + cos(kt)

Question 18

Suppose the position vector r(t) of an object moving in 3-space and the corresponding velocity and acceleration vectors v(t) and a(t) satisfy
(i) a(t) is parallel to r(t) for all t, and
(ii) r(0) × v(0) = c (a nonzero constant vector).
Find r(t) × v(t) and c . r(t) for all t.

Question 19

Assuming u(t) has continuous derivatives of all required orders, simplify the expression .

A) . + u(t) .
B) .
C) u(t) .
D) 0
E) . + u(t) .

Question 20

Assuming u(t) has continuous derivatives of all required orders, simplify the expression .

A) × + u(t) ×
B) ×
C) u(t) ×
D) 0
E) × + u(t) ×

Question 21

If u and v are differentiable functions of t, then

Question 22

If r(t) , v(t) and a(t) are the position, velocity, and acceleration of a moving particle at any time t, respectively, and if the speed at any time t is a constant , then the velocity and acceleration vectors are perpendicular.

Question 23

A particle moves around the elliptical cylinder + = 1 in such a way that its position at time t is Find the maximum and minimum values of both its speed and its magnitude of its acceleration .

A) max = 3, min = 2; max = 3, min = 2
B) max = 5, min = 2 ; max = 3, min = 2
C) max = 5, min = 2 ; max = 5, min = 2
D) max = 3, min = 2; max = 3, min = 0
E) max = 5, min = 2 ; max = 3, min = 0

Question 24

A ball of ice having mass 100 g at time t = 0 s is melting, and therefore losing mass, at a steady rate of 1 g/s. The ball has velocity i + 2 j at time t = 0 and is subject to a constant force F = 3 i thereafter. What is its velocity after 1 minute?

A) 10 i + 5 j
B) 10 i + 10 j
C) 4 i + 5 j
D) 4 i + 8 j
E) 10 i + 8 j

Question 25

Solve the initial-value problem = k × r with initial condition r(0) = i + 2j + 3k. Describe the solution curve.

A) r(t) = i + j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.
B) r(t) = i - j + 3 k,a circle with centre (0, 0, 3) and radius in the plane z = 3.
C) r(t) = i - j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.
D) r(t) = i + j + 3 k,a circle with centre (0, 0, 3) and radius in the plane z = 3.
E) r(t) = i + j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.

Question 26

You are at the origin in the xy-plane. At time t = 0 an incoming missile is at position (1000, 500) and has velocity -30 i + 3 j. (All distances are in metres and time is measured in seconds.) At that instant you fire an anti-missile missile to intercept the incoming missile. If your missile has an initial speed of 100 m/s, at what angle of elevation above the horizontal should you fire your missile to ensure that it intercepts the incoming missile? Assume that gravity is the only force acting on the projectiles.

A) 0.42535 radians
B) 0.52535 radians
C) 0.62535 radians
D) 0.72535 radians
E) 0.56535 radians

Question 27

A rocket is fired vertically in a vacuum (free space where the gravitational field is negligible).During the burning process, the exhaust gases are ejected at a constant rate of 1000 kg/s and at a constant velocity with magnitude 400 m/s relative to the rocket.Let M be the total initial mass of the rocket, and assume the rocket starts motion from rest.In order to accelerate to the speed of 800 m/s, the rocket has to burn P % of the total initial mass M as a fuel. Assuming there is sufficient amount of fuel on board, the value of P is equal to:

A) 1 -
B) 100
C) 100
D)
E) 100 -

Question 28

A rocket with mass 60,000 kilogram (kg), which includes 30,000 kg of fuel mixture, is fired vertically in a vacuum (free space where the gravitational field is negligible).During the burning process, the exhaust gases are ejected at a constant rate of 1000 kg/s and at a constant velocity with magnitude 400 m/s relative to the rocket.If the rocket was initially at rest, find its speed after 50 seconds.

A) 400 ln(3) m/s
B) 400 ln m/s
C) 400 m/s
D) 400 ln (2) m/s
E) 400 ln (6) m/s

Question 29

Which of the following parametrize the circle + - 4y = 0 with clockwise orientation?
(a) r = (2 cos $\theta$ ) i + (2 + 2 sin $\theta$ ) j
(b) r = (2 cos $\theta$ ) i - (2 + 2 sin $\theta$ ) j
(c) r = (2 cos $\theta$ ) i + (2 - 2 sin $\theta$ ) j
(d) r = (2 sin $\theta$ ) i + (2 + 2 cos $\theta$ ) j

A) only (a)
B) only (b)
C) only (c)
D) only (d)
E) both (a) and (b)
F) both (c) and (d)
G) none of the above

Question 30

Find a parametrization of the ellipse in which the plane z = 3y intersects the cylinder + = 4, using the polar angle $\theta$ in the xy-plane as the parameter, [0, 2 $\pi$ ] as parameter interval, and ensuring that the ellipse is oriented counterclockwise as viewed from high on the z-axis.

A) r = 2 cos( $\theta$ ) i + 2 sin( $\theta$ ) j + 6 sin( $\theta$ ) k
B) r = 2 cos( $\theta$ ) i - 2 sin( $\theta$ ) j + 6 sin( $\theta$ ) k
C) r = 2 cos( $\theta$ ) i + 2 sin( $\theta$ ) j - 6 sin( $\theta$ ) k
D) r = 2 cos( $\theta$ ) i - 2 sin( $\theta$ ) j - 6 sin( $\theta$ ) k
E) r = 2 cos( $\theta$ ) i + 2 sin( $\theta$ ) j + 3 sin( $\theta$ ) k

Question 31

Parametrize the parabola in which the plane z = 1 + y intersects the cone + = using x as the parameter.

A) r = x i + j + k, - $\infty$ < x < $\infty$
B) r = x i + j + k, - $\infty$ < x < $\infty$
C) r = x i + j + k, - $\infty$ < x < $\infty$
D) r = x i + j + k, - $\infty$ < x < $\infty$
E) r = x i + j + k, - $\infty$ < x < $\infty$

Question 32

Find the length of the arc r = u i + j, 0 $\le$ u $\le$ .

A) units
B) units
C) units
D) units
E) units

Question 33

Find the perimeter of the astroid curve x = 2 $\theta$ , y = 2 $\theta$ .

A) 16 units
B) 6 $\pi$ units
C) 3 $\pi$ units
D) 12 units
E) 9 $\pi$ units

Question 34

Find the arc length of the arc r = (ln (cos $\theta$ ) ) i + $\theta$ j, - $\le$ $\theta$ $\le$ 0.

A) units
B) ln (2 + ) units
C) 2 units
D) units
E) units

Question 35

Find the length of the curve r(t) = ( sin t) i + ( cos t) j + 2t k for 0 $\le$ t $\le$ 2 $\pi$ .

A) + 2 $\pi$ units
B) + 4 $\pi$ units
C) + 4 $\pi$ units
D) + 2 $\pi$ units
E) + 2 $\pi$ units

Question 36

Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - ) k, 1 $\le$ t $\le$ 2.

A) 1 + 2ln(4) units
B) unit
C) 2 units
D) 1 unit
E) 3 units

Question 37

Parametrize the curve of intersection of the cylinder y = 1 - and the plane z = - x using as parameter the slope m = of the curve of intersection of the cylinder with the xy-plane.

A) x = - i + j + m k
B) x = - m i + j + m k
C) x = - i + j + k
D) x = - i + j - k
E) x = i + j - k

Question 38

Find a parametric representation of the curve of intersection of the paraboloid z = x² + y² and the plane 8x - 4y - z - 11 = 0.

A) r = (3 - 4cos(t)) i + (3 + 2sin(t)) j + (1 - 32cos(t) - 8sin(t)) k, 0 $\le$ t $\le$ 2 $\pi$
B) r = (3 + 4cos(t)) i + (3 - 2sin(t)) j + (1 + 32cos(t) + 8sin(t)) k, 0 $\le$ t $\le$ 2 $\pi$
C) r = (4 + 3cos(t)) i + (-2 + 3sin(t)) j + (24cos(t) - 12sin(t) - 29) k, 0 $\le$ t $\le$ 2 $\pi$
D) r = (-4 + 3cos(t)) i + (2 + 3sin(t)) j + (24cos(t) - 12sin(t) - 51) k, 0 $\le$ t $\le$ 2 $\pi$
E) r = (-3 + 4cos(t)) i + (-3 - 2sin(t)) j + (32cos(t) + 8sin(t) - 23) k, 0 $\le$ t $\le$ 2 $\pi$

Question 39

Reparametrize the curve r = cos t) i + ( sin t) j + k in terms of arc length s measured from the point where t = 0.

A) r = cos i + sin j + k
B) r = cos i + sin j + k
C) r = cos i + sin j + k
D) r = cos i + sin j + k
E) r = i + j + k

Question 40

A parametric representation of the curve of intersection of the two surfaces 4x² + y² + z² = 8 and is given by which of the following vector equations?

A) r = 2cos(t) i + sin(t) j - 2 k, 0 $\le$ t $\le$ 2 $\pi$
B) r = 2cos(t) i + sin(t) j + 2 k, 0 $\le$ t $\le$ 2 $\pi$
C) r = cos(t) i + 2sin(t) j +2 k, 0 $\le$ t $\le$ $\pi$
D) r = cos(t) i + 2sin(t) j - 2 k, 0 $\le$ t $\le$ 2 $\pi$
E) r = 2cos(t) i + sin(t) j - 2 k, 0 $\le$ t $\le$ $\pi$

Question 41

A recording tape 0.01 cm thick is wound around a reel whose inner radius is 1 cm and outer radius is 4 cm. How much tape is required to fill the reel?

A) 1885 cm
B) 1178 cm
C) 4712 cm
D) 2827 cm
E) 5238 cm

Question 42

Let , , and be the unit tangent, the principal unit normal, and the unit binormal, respectively.Which of the following equations is correct?

A) = ×
B) = ×
C) = .
D) = ×
E) = +

Question 43

Find the curvature of a circle with radius a.

A)
B)
C) -
D) a
E)

Question 44

Find the radius of curvature of the plane curve 4x² + 4y² + 16x - 12y -11 = 0.

A) 6
B) 3
C) 2
D)
E)

Question 45

Find and for the circular motion described by r(t) = (cos bt) i + (sin bt) j.

A) = -(sin bt) i + (cos bt) j and = -(cos bt) i - (sin bt) j
B) = (sin bt) i + (cos bt) j and = (cos bt) i - (sin bt) j
C) = -(sin bt) i + (cos bt) j and = -(cos bt) i + (sin bt) j
D) = -(sin bt) i - (cos bt) j and = -(cos bt) i + (sin bt) j
E) = (sin bt) i + (cos bt) j and = -(cos bt) i + (sin bt) j

Question 46

For what positive value of the constant c is the curve r = (3 cos cs) i + (3 sin cs) j + 4cs k parametrized in terms of arc length s? What is the curvature (s) and the torsion (s) for this curve? What kind of curve is it?

A) c = , 11ee7b4a_c535_8783_ae82_218a633b3a19_TB9661_11 = , 11ee7b4a_f17d_2194_ae82_9be00d89db23_TB9661_11 = 0, circle
B) c = , 11ee7b4a_c535_8783_ae82_218a633b3a19_TB9661_11 = , 11ee7b4a_f17d_2194_ae82_9be00d89db23_TB9661_11 = , circular helix
C) c = , 11ee7b4a_c535_8783_ae82_218a633b3a19_TB9661_11 = , 11ee7b4a_f17d_2194_ae82_9be00d89db23_TB9661_11= , circular helix
D) c = , 11ee7b4a_c535_8783_ae82_218a633b3a19_TB9661_11 = , 11ee7b4a_f17d_2194_ae82_9be00d89db23_TB9661_11 = , circular helix
E) c = , 11ee7b4a_c535_8783_ae82_218a633b3a19_TB9661_11 = , 11ee7b4a_f17d_2194_ae82_9be00d89db23_TB9661_11 = , circular helix

Question 47

Let : I be the unit speed vector 11ee7b18_ef83_6016_ae82_a1fef198316c_TB9661_11 (s) = . Compute the curvature and torsion of https://d2lvgg3v3hfg70.cloudfront.net/TB9661/11ee7b18_ef83_6016_ae82_a1fef198316c_TB9661_11.

A) Curvature is and torsion is
B) Curvature is and torsion is
C) Curvature is and torsion is
D) Curvature is and torsion is
E) Curvature is and torsion is -

Question 48

Let r = r(s) be a curve parametrized in terms of arc length, let (s) and (s) be the curvature and torsion, and let { , , } be the Frenet frame for the curve. Suppose that = 0 and = 0 for Calculate , , and in terms of , , , , and .

A)
B)
C)
D)
E)

Question 49

A curve with constant curvature and constant torsion is a circular helix. What is the radius of the cylinder on which the helix is wound if the curvature is 1 and the torsion is 2?

A)
B)
C)
D) 5
E) 1

Question 50

A curve can have constant curvature = 2 and constant torsion = 0.

Question 51

A curve can have constant curvature = 0 and constant torsion = 2.

Question 52

The curve r = r(s) is a straight line if and only if (s) = 0 for all s.

Question 53

A curve r(s) parametrized in terms of arc length s is traced at unit speed (thus, =1).

Question 54

Find the curvature of the parabola y = x² at the point (0, 0).

A) 2
B) 1
C) 4
D)
E)

Question 55

Find the curvature of the hyperbola xy = 1 at (1, 1).

A)
B) 2
C)
D)
E) 2

Question 56

Find the maximum and minimum values of the curvature of the ellipse r = (3 cos t) i + (2 sin t) j.

A) max , min
B) max , min
C) max , min
D) max 2, min
E) max , min

Question 57

At what value of x is the radius of curvature of y = e^x smallest?

A) -
B)
C) -
D)
E)

Question 58

Find and for the plane curve r(t) = (2t + 3) i + (5 - ) j.

A) = i + j, = - i + j
B) = i + j, = i - j
C) = i - j, = i + j
D) = i - j, = - i - j
E) = i - j, = - i - j

Question 59

The acceleration, the unit tangent, and the principal unit normal of a moving particle in space at a particular time are respectively given by
a = 12 i + 3 j -12 k , = ( 2 i + j - 2 k ), and = ( i -4 j - k )
The tangential component of the acceleration is equal to:

A) 2 ( i - j + k )
B) 17
C) 8 i + j + 8 k
D) 2
E) 0

Question 60

Find the radius of curvature of the curve r = (2 cos t) i + (2 sin t) j + (2 sin t) k.

A) = 2
B) 11efb6fe_8167_da73_bb63_af6e34405a3b_TB9661_00 = 2
C) 11efb6fe_8167_da73_bb63_af6e34405a3b_TB9661_00 =
D) 11efb6fe_8167_da73_bb63_af6e34405a3b_TB9661_00 =
E)11efb6fe_8167_da73_bb63_af6e34405a3b_TB9661_00 =

Question 61

Find the curvature of r = ( t) i + ( t) j at t = .

A)
B) -
C)
D)
E)

Question 62

Let C be the space curve given by r(t) = (t - ) i + 3t j + (2t - ) k. The unit binormal to curve C at t = 0 is given by

A) - i - j + k
B) - i + j + k
C) - i + j + k
D) - i - j + k
E) - k

Question 63

Find the curvature and the torsion of the curve of intersection of the surfaces x² + y² + z² = 18 and at an arbitrary point (x, y, z).

Question 64

Find the point on the curve r(t) = (5 sin t) i + (5 cos t) j + 12t k at a distance 26 $\pi$ units along the curve from the point (0, 5, 0) when t > 0 corresponds to the direction of increasing arc length.

A) (-5, 0, 2 $\pi$ )
B) (5, 0, 12 $\pi$ )
C) (0, 5, 26 $\pi$ )
D) (0, 5, 24 $\pi$ )
E) (5, 0,12 $\pi$ )

Question 65

Find the radius of curvature of r = .

A)
B)
C)
D)
E)

Question 66

Find the Frenet frame for the curve r = .

A) (t) = , (t) = , (t) =
B) (t) = , (t) = , (t) =
C) (t) = , (t) = , (t) =
D) (t) = , (t) = , (t) =
E) (t) = , (t) = , (t) =

Question 67

Find an equation for the osculating circle of the curve r = t i + (sin t) j at the point ( , 1).

A) + = 4
B) + = 2
C) + = 1
D) + =
E) + = 2

Question 68

Find the radius of curvature of the cycloid r = a ( $\theta$ - sin $\theta$ ) i + a(1 - cos $\theta$ ) j at the point $\theta$ = $\pi$ . Assume that a > 0.

A) 4a
B)
C) 2a
D)
E) a

Question 69

The speed and the magnitude of the acceleration of a moving particle at some point P are 3 units and 4 units, respectively. If the angle between the velocity and acceleration vectors at the point P is 30°, determine the radius of curvature of the particle's trajectory at P.

A)
B) 4.5
C)
D)
E) 2.25
F)

Question 70

Find the tangential and normal components of the acceleration of a particle moving so that at time t its position is r(t) = ( cos t) i + ( sin t) j.

A) = , =
B) = 2 , =
C) = , = 2
D) = 2 , =
E) = , =

Question 71

Find the evolute (t) of the curve r(t) = ( cos t) i + ( sin t) j + 2k.

A) = - ( sin t) i + ( cos t) j + 2k
B) = ( sin t) i + ( cos t) j + 2k
C) = ( sin t) i - ( cos t) j
D) = - ( sin t) i - ( cos t) j
E) = - ( sin t) i - ( cos t) j + k

Question 72

Find the evolute (t) of the curve r(t) = (2 cos t) i + (2 sin t) j + 3t k.

A) = - sin t i - cos t j + 3t k
B)
C)
D)
E)

Question 73

Find a polynomial f(x) of lowest possible degree such that the curve y = f(x), 0 $\le$ x $\le$ 1, can be used to join the straight line segments y = 0, x $\le$ 0, and y = 1, x $\ge$ 1, to form a curve along which a particle can travel at constant speed without experiencing discontinuous acceleration.

A) f(x) = 10 - 15 + 6
B) f(x) = 4 - 7 + 4
C) f(x) = 2 - 3 + 2
D) f(x) = 18 - 30 + 13
E) f(x) = - 3 + 2

Question 74

A curve with nonzero curvature lies in a plane if and only if the torsion of the curve is identically zero.

Question 75

A frictionless highway turn has a constant curvature 1.96 × and is banked at an angle $\theta$ = (0.2). What is the maximum safe speed for the turn in m/s?
You may assume the gravitational acceleration g = 9.8 m/ .

A) 100 m/s
B) 19.6 m/s
C) 20 m/s
D) 25 m/s
E) 10 m/s

Question 76

A frictionless road turn is approximately circular of radius 50 metres and is designed for a maximum safe speed of 10 m/s. Determine the banking angle of the turn to the nearest degree.You may assume the gravitational acceleration g = 9.8 m/s².

Question 77

The period of the moon's orbit around the Earth is approximately 27.32 days, and the semi-major axis of its orbit is about 385 000 km. Find the radius and location of the circular orbit of a geosynchronous satellite (i.e., one that remains above the same point on the Earth's surface).

A) 69 742 km, in the equatorial plane
B) 35 016 km, in the equatorial plane
C) 42 443 km, in the equatorial plane
D) 45 229 km, in a polar plane
E) none of the above

Question 78

The distance from the moon to the Earth at perigee is about 356 000 km and its distance at apogee is about 406 700 km. Find the eccentricity of the moon's orbit.

A) 0.066
B) 0.058
C) 0.051
D) 0.043
E) none of the above

Question 79

The angular velocity of a certain comet at perihelion is 10 times its angular velocity at aphelion. Find the eccentricity of the comet's orbit.

A)
B)
C)
D)
E) none of the above

Question 80

Find the radial and transverse components of the acceleration of a particle moving with constant speed v along the polar curve r = . Express your answer s as functions of $\theta$ .

A) = - , = -
B) = - , =
C) = - , =
D) = , =
E) none of the above