The James Webb Space Telescope (JWST) is a now space telescope currently under construction. The JWST will orbit the Sun in an orbit roughly 1.5 million fm from the earth, in a position with the Earth between the telescope and the Sun. The primary mirror of the JWST will consist of 18 hexagonally shaped segments that when fitted together will be 6.5 m in diameter (over 2.5 times the diameter of the Hubble). With a larger mirror, the JWST will be able to collect more light to “see” deeper into space. The primary mirror when pieced together will function as a parabolic mirror with a focal length of 131.4 m. Suppose that a coordinate system is chosen with ( 0 , 0 ) at the center of a cross section of the primary mirror. Write an equation of the parabolic cross section of the mirror for − 3.25 ≤ x ≤ 3.25 . Assume that x and y are measured in meters.
The James Webb Space Telescope (JWST) is a now space telescope currently under construction. The JWST will orbit the Sun in an orbit roughly 1.5 million fm from the earth, in a position with the Earth between the telescope and the Sun. The primary mirror of the JWST will consist of 18 hexagonally shaped segments that when fitted together will be 6.5 m in diameter (over 2.5 times the diameter of the Hubble). With a larger mirror, the JWST will be able to collect more light to “see” deeper into space. The primary mirror when pieced together will function as a parabolic mirror with a focal length of 131.4 m. Suppose that a coordinate system is chosen with ( 0 , 0 ) at the center of a cross section of the primary mirror. Write an equation of the parabolic cross section of the mirror for − 3.25 ≤ x ≤ 3.25 . Assume that x and y are measured in meters.
Solution Summary: The author calculates the parabolic cross section of the mirror with focal length of 131.4m.
The James Webb Space Telescope (JWST) is a now space telescope currently under construction. The JWST will orbit the Sun in an orbit roughly 1.5 million fm from the earth, in a position with the Earth between the telescope and the Sun. The primary mirror of the JWST will consist of 18 hexagonally shaped segments that when fitted together will be 6.5 m in diameter (over 2.5 times the diameter of the Hubble). With a larger mirror, the JWST will be able to collect more light to “see” deeper into space.
The primary mirror when pieced together will function as a parabolic mirror with a focal length of 131.4 m. Suppose that a coordinate system is chosen with
(
0
,
0
)
at the center of a cross section of the primary mirror. Write an equation of the parabolic cross section of the mirror for
−
3.25
≤
x
≤
3.25
. Assume that x and y are measured in meters.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
A star rotates in a circular orbit about the center of its galaxy. The radius of the orbit is 3.7 x 1020 m, and the angular speed of the star is 9.3 x 10-15 rad/s. How long (in years) does it take for the star to make one revolution around the center?
In a distance star system a planet is 9 AU (astronomical unit) away from the sun, and has a circular orbit with period 12 earth years. The planet has a moon distanced 0.5 AU away, which circles the planet in a circular orbit also with period 12 earth years. Suppose the orbital planes of the planet around the sun and the moon around the planet coincide, and their motions are both counterclockwise. At midnight March 1 2021, the positions of the sun, planet, and moon are observed to be (0, 0), (9, 0), (9, 0.5) on their common orbital plane.
(a) Determine the position (xP (t), yP (t)) of the planet, t years after midnight March 1 2021.
(b) Determine the position (xM (t), yM (t)) of the moon, t years after midnight March 1 2021. Express your
answer in the form
xM (t) = A cos(kt + α) + xP (t) yM (t) = A sin(kt + α) + yP (t)
where A,k > 0 and α ∈ (−π,π].
(c) Simplify your expressions for xM (t) and yM (t) to the form
xM (t) = B cos(kt + β) yM (t) = C sin(kt + γ)
where B, C, k >…
Karina has an antique one-speed bicycle that has two gears connected by a chain.
The front gear has a radius of 7 inches and the rear gear has a radius of 1.5 inches.
The chain and teeth of each gear have the same linear speed. The tires have a
diameter of 30 inches.
Karina is peddling at a rate of 15 rotations per minute, meaning the front gear is
turning at that speed. What is the exact angular speed of the front gear?
radians per minute
Enter the exact answer; use pi for .
What is the exact linear speed of the chain?
inches per minute
Enter the exact answer; use pi for T.
What is the exact rotational speed of the rear gear?
rotations per minute
Enter the exact answer; use pi for T.
The rear wheel has the same angular velocity as the rear gear. How fast is Karina
travelling?
miles per hour
Round to the nearest integer. One inch per minute =
1
miles per hour.
1056
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