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
Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be \[\sqrt{8}\] T.
Answer the following question.
State Kepler’s law of equal areas.
Answer the following question.
State Kepler’s law of the period.
The orbit of a planet revolving around a star is ______.
State Kepler’s laws.
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
A planet revolves in an elliptical orbit around the sun. The semi-major and minor axes are a and b, then the time period is given by:
In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They ______.
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.
The centre of mass of an extended body on the surface of the earth and its centre of gravity ______.
- are always at the same point for any size of the body.
- are always at the same point only for spherical bodies.
- can never be at the same point.
- is close to each other for objects, say of sizes less than 100 m.
- both can change if the object is taken deep inside the earth.
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?
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:
What is at one focus of the elliptical orbit of a planet?
Two identical particles each of mass ‘m’ go round a circle of radius a under the action of their mutual gravitational attraction. The angular speed of each particle will be ______
