For a planet orbiting the sun, r p is the distance from the sun to the perihelion and r a is the distance from the sun to the aphelion. What is a planet’s orbital eccentricity if r p is equal to 0.45r a?
A. 1.45
B. 0.55
C. 0.38
D. 2.64

Where is the sun located on Earth’s elliptical orbit?
A. at the center of the two foci
B. at a focus
C. at the aphelium
D. at the perihelium

A satellite orbiting Earth at an orbital radius r has a velocity v. Which represents the velocity if the satellite is moved to an orbital radius of 4r?
A. 1/4v
B. 4v
C. 1/2v
D. 2v

1) 0.38 (C)

2) At a focus (B)

3) (1/2)v (C)

Explanation:

1) Eccentricity, e = (Ra - Rp)/(Ra +Rp)

Let r a = Ra, rp = Rp

Rp = 0.45Ra

e = (Ra - 0.45Ra)/(Ra + 0.45Ra)

e= 0.55Ra/1.45Ra

e = 55/145

e= 0.38

2) The sun is located at a focus on Earth’s elliptical orbit:

The different location in the Earth’s orbit of the Sun is determined by the perihelion and aphelion.

The perihelion is the point on earth's orbit closest to the sun while aphelion is the point on earth's orbit that is far away from the sun.

The Sun tends a little bit towards one side at a point known as the focus of an ellipse.

This out of alignment causes the planet to move either closer to or further away from the Sun every orbit.

3) initial velocity = v

radius = r

v = √(GM/r)

v = (√GM/√r)

From the expression above, Velocity is directly proportional to the inverse of square root of radius

v = 1/√r

If radius become 4r, the new velocity (V):

V = 1/√4r

V = 1/(2√r) = 1/2(1/√r)

Recall, v = 1/√r

V = (1/2)(v)

V = v/2

Hence velocity decreases as r increases.

New velocity = v/2

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.