Advertisements
Advertisements
प्रश्न
A satellite is in an elliptic orbit around the earth with aphelion of 6R and perihelion of 2 R where R= 6400 km is the radius of the earth. Find eccentricity of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius 6R ?
[G = 6.67 × 10–11 SI units and M = 6 × 1024 kg]
Advertisements
उत्तर
According to the diagram,
rp = radius of perigee = 2R
ra = radius of perigee = 6R
a = semi-major axis of the ellipse
Hence, we can write
`r_a = a(1 + e) = 6R`
`r_p = a(1 - e) = 2R`
`(a(1 + e))/(a(1 - e)) = (6R)/(2R)` = 3
By solving, we get eccentricity `e = 1/2`
If va and vp are the velocities of the satellite (of mass m) at aphelion and perihelion respectively, then by conservation of angular momentum
`L_("at perigee") = L_("at apogee")`
∴ `mv_pr_p = mv_ar_a`
∴ `v_a/v_p = r_p/r_a = 1/3`
Applying conservation of energy,
The energy at perigee = Energy at apogee
`1/2 mv_p^2 - (GMm)/r_p = 1/2 mv_a^2 - (GMm)/r_a`
Where M is the mass of the earth
∴ `v_p^2 (1 - 1/9) = - 2GM (1/r_a - 1/r_p)`
= `2GM(1/r_p - 1/r_a)` .....(By putting `v_a = v_p/3`)
`v_p = [2GM(1/r_p - 1/r_a)]^(1/2)/([1 - (v_a/v_p)^2]^(1/2)`
= `[((2GM)/R (1/2 - 1/6))/((1 - 1/9))]^(1/2)`
= `((2/3)/(8/9) (GM)/R)^(1/2)`
= `sqrt(3/4 (GM)/R)`
= 6.85 km/s
vp = 6.885 km/s, va = 2.28 km/s
For circular orbit of radius r,
vc = orbital velocity = `sqrt((GM)/r)`
For r = 6R, vc = `sqrt((GM)/(6R)` = 3.23 km/s.
APPEARS IN
संबंधित प्रश्न
A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50 ×108 km away from the sun?
Identify the law shown in the figure and state three respective laws.
Answer the following question in detail.
State Kepler’s three laws of planetary motion.
The third law of Kepler is also known as the Law of ______.
If the distance between the sun and the earth is made three times, then attraction between the two will ______
The earth moves around the sun in an elliptical orbit as shown in the figure. The ratio, `"OA"/"OB"` = x. The ratio of the speed of the earth at Band at A is ______.
To verify Kepler's third law graphically four students plotted graphs. Student A plotted a graph of T (period of revolution of planets) versus r (average distance of planets from the sun) and found the plot is straight line with slope 1.85. Student B plotted a graph of T2 v/s r3 and found the plot is straight line with slope 1.39 and negative Y-intercept. Student C plotted graph of log T v/s log r and found the plot is straight line with slope 1.5. Student D plotted graph of log T v/s log r and found the plot is straight line with slope 0.67 and with negative X-intercept. The correct graph is of student
If the sun and the planets carried huge amounts of opposite charges ______.
- all three of Kepler’s laws would still be valid.
- only the third law will be valid.
- the second law will not change.
- the first law will still be valid.
If the sun and the planets carried huge amounts of opposite charges ______.
- all three of Kepler’s laws would still be valid.
- only the third law will be valid.
- the second law will not change.
- the first law will still be valid.
Supposing Newton’s law of gravitation for gravitation forces F1 and F2 between two masses m1 and m2 at positions r1 and r2 read F1 = – F2 = `- r_12/r_12^3 GM_0^2 ((m_1m_2)/M_0^2)^n` where M0 is a constant of dimension of mass r12 = r1 – r2 and n is a number. in such a case.
- the acceleration due to gravity on earth will be different for different objects.
- none of the three laws of Kepler will be valid.
- only the third law will become invalid.
- for n negative, an object lighter than water will sink in water.
Draw areal velocity versus time graph for mars.
What is the direction of areal velocity of the earth around the sun?
Out of aphelion and perihelion, where is the speed of the earth more and why?
The maximum and minimum distances of a comet from the Sun are 1.6 × 1012 m and 8.0 × 1010 m respectively. If the speed of the comet at the nearest point is 6 × 104 ms-1, the speed at the farthest point is ______.
A planet revolving in an elliptical orbit has:
- a constant velocity of revolution.
- has the least velocity when it is nearest to the sun.
- its areal velocity is directly proportional to its velocity.
- areal velocity is inversely proportional to its velocity.
- to follow a trajectory such that the areal velocity is constant.
Choose the correct answer from the options given below:
How can an ellipse be drawn using pins and thread?
When is a planet moving fastest in its orbit?
