A planet Mass M is circling a star in an elliptical orbit as shown figure. if its velocity at a point A is 60000m/s,and rb=8ra what is its velocity at point b?
answer pls
To find the velocity of the planet at point B, we can use the principle of conservation of angular momentum. The angular momentum of the planet is constant as it moves in its elliptical orbit.
The formula for angular momentum is given by:
L = mvr
Where:
L = angular momentum
m = mass of the planet
v = velocity of the planet
r = distance from the planet to the center of the star
Since the mass of the planet, M, is mentioned, we can assume it is the mass referred to in the equation. Therefore, we can rewrite the equation as:
L = Mvr
Since angular momentum is constant, we have:
L at point A = L at point B
Let's calculate the angular momentum at point A and B:
L at point A = M * vA * rA
L at point B = M * vB * rB
Given:
vA = 60000 m/s
rb = 8 * ra
Substituting the values into the equation:
M * vA * rA = M * vB * rB
Since we want to find the velocity at point B, let's rearrange the equation:
vB = (vA * rA) / rB
Now, we need to find rA and rB.
The orbital parameters given are ra and rb, which represent the distance of the planet from the star at points A and B, respectively. The ratio is given as rb = 8ra. So, we can substitute this relationship into the equation:
rB = 8 * ra
Substituting the values:
vB = (vA * rA) / (8 * ra)
Simplifying the equation further:
vB = vA / 8
Finally, substitute the given value of vA = 60000 m/s into the equation:
vB = 60000 m/s / 8
vB = 7500 m/s
Therefore, the velocity of the planet at point B is 7500 m/s.
To determine the velocity at point B, we can use the conservation of angular momentum principle, as the planet is orbiting the star.
1. The conservation of angular momentum states that the angular momentum of a planet remains constant throughout its orbit. It is given by the equation:
L = mvr,
where L is the angular momentum, m is the mass of the planet, v is the velocity, and r is the distance from the planet to the star.
2. Since the mass of the planet (M) is given, we can express the angular momentum at point A:
L(A) = M * v(A) * r(A),
where v(A) is the velocity at point A and r(A) is the distance from the planet to the star at point A.
3. Now, we need to relate the distances at points A and B. The problem states that rb = 8ra, where ra is the distance from the planet to the star at point A, and rb is the distance at point B.
4. Using the relationship between the distances, we can express the angular momentum at point B:
L(B) = M * v(B) * r(B).
5. Since angular momentum is conserved, we can equate the angular momentum at points A and B:
L(A) = L(B).
M * v(A) * r(A) = M * v(B) * r(B).
6. Rearranging the equation, we find:
v(B) = v(A) * (r(A) / r(B)).
7. Substitute the given values:
v(B) = 60000 m/s * (r(A) / r(B)).
8. Since rb = 8ra, we can rewrite the equation as:
v(B) = 60000 m/s * (ra / rb).
9. Plug in the values of ra and rb:
v(B) = 60000 m/s * (ra / 8ra) = 60000 m/s / 8 = 7500 m/s.
Therefore, the velocity of the planet at point B is 7500 m/s.