359958
A planet moving along an elliptical orbit is close to the sun at a distance \(r_{1}\) and farthest away at a distance of \(r_{2}\). If \(v_{1}\) and \(v_{2}\) are the linear velocities at these points respectively. Then the ratio \(\frac{{{v_1}}}{{{v_2}}}\) is
1 \(\dfrac{r_{2}}{r_{1}}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
Explanation:
From conservation of angular momentum of the planet of mass \(m\). \(\frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}\) [angular momentum is constant]
PHXI08:GRAVITATION
359959
The ratio of \(K E\) of a planet at the points 1 and 2 is
1 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\dfrac{r_{2}}{r_{1}}\)
Explanation:
\(v_{1} r_{1}=v_{2} r_{2} \Rightarrow \dfrac{v_{1}}{v_{2}}=\dfrac{r_{2}}{r_{1}}\) So, \(\dfrac{K_{1}}{K_{2}}=\left(\dfrac{v_{1}}{v_{2}}\right)^{2}=\dfrac{r_{2}^{2}}{r_{1}^{2}}\).
PHXI08:GRAVITATION
359960
If ' \(A\) ' is areal velocity of a planet of mass \(M\), its angular momentum is
1 \(M / A\)
2 \(2 M A\)
3 \(A^{2} M\)
4 \(A M^{2}\)
Explanation:
\(\dfrac{d A}{d t} \rightarrow\) Areal velocity of a planet revolving around sun. \(\dfrac{d A}{d t}=\dfrac{L}{2 M}, L \rightarrow\) Angular momentum of planet. \( \Rightarrow L = 2M\frac{{dA}}{{dt}} = 2\,MA\).
PHXI08:GRAVITATION
359961
The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
1 \(\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\)
2 \(\dfrac{r_{1}+r_{2}}{3}\)
3 \(\dfrac{r_{1}+r_{2}}{4}\)
4 \(\dfrac{r_{1} r_{2}}{r_{1}+r_{2}}\)
Explanation:
The earth moves around the sun in elliptical path as shown in the figure. So by using the properties of ellipse \(r_{1}=(1+e) a \text { and } r_{2}=(1-e) a\) where, \(a=\) length of semi-major axis \(b=\) length of semi-major axis \(e=\) eccentricity Given that \(\Rightarrow a=\dfrac{r_{1}+r_{2}}{2}\) and \(r_{1} r_{2}=\left(1-e^{2}\right) a^{2}\) Now, required distance \(=\) semi latusrectum \(\begin{aligned}F_{1} P & =\dfrac{b^{2}}{a} \\& =\dfrac{a^{2}\left(1-e^{2}\right)}{a}=\dfrac{r_{1} r_{2}}{\left(r_{1}+r_{2}\right) / 2}=\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\end{aligned}\)
PHXI08:GRAVITATION
359962
Average distance of the earth from the sun is \(L_{1}\). If one year of earth \(=D\) days, one year of another planet whose average distance from the sum is \(L_{2}\) will be
359958
A planet moving along an elliptical orbit is close to the sun at a distance \(r_{1}\) and farthest away at a distance of \(r_{2}\). If \(v_{1}\) and \(v_{2}\) are the linear velocities at these points respectively. Then the ratio \(\frac{{{v_1}}}{{{v_2}}}\) is
1 \(\dfrac{r_{2}}{r_{1}}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
Explanation:
From conservation of angular momentum of the planet of mass \(m\). \(\frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}\) [angular momentum is constant]
PHXI08:GRAVITATION
359959
The ratio of \(K E\) of a planet at the points 1 and 2 is
1 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\dfrac{r_{2}}{r_{1}}\)
Explanation:
\(v_{1} r_{1}=v_{2} r_{2} \Rightarrow \dfrac{v_{1}}{v_{2}}=\dfrac{r_{2}}{r_{1}}\) So, \(\dfrac{K_{1}}{K_{2}}=\left(\dfrac{v_{1}}{v_{2}}\right)^{2}=\dfrac{r_{2}^{2}}{r_{1}^{2}}\).
PHXI08:GRAVITATION
359960
If ' \(A\) ' is areal velocity of a planet of mass \(M\), its angular momentum is
1 \(M / A\)
2 \(2 M A\)
3 \(A^{2} M\)
4 \(A M^{2}\)
Explanation:
\(\dfrac{d A}{d t} \rightarrow\) Areal velocity of a planet revolving around sun. \(\dfrac{d A}{d t}=\dfrac{L}{2 M}, L \rightarrow\) Angular momentum of planet. \( \Rightarrow L = 2M\frac{{dA}}{{dt}} = 2\,MA\).
PHXI08:GRAVITATION
359961
The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
1 \(\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\)
2 \(\dfrac{r_{1}+r_{2}}{3}\)
3 \(\dfrac{r_{1}+r_{2}}{4}\)
4 \(\dfrac{r_{1} r_{2}}{r_{1}+r_{2}}\)
Explanation:
The earth moves around the sun in elliptical path as shown in the figure. So by using the properties of ellipse \(r_{1}=(1+e) a \text { and } r_{2}=(1-e) a\) where, \(a=\) length of semi-major axis \(b=\) length of semi-major axis \(e=\) eccentricity Given that \(\Rightarrow a=\dfrac{r_{1}+r_{2}}{2}\) and \(r_{1} r_{2}=\left(1-e^{2}\right) a^{2}\) Now, required distance \(=\) semi latusrectum \(\begin{aligned}F_{1} P & =\dfrac{b^{2}}{a} \\& =\dfrac{a^{2}\left(1-e^{2}\right)}{a}=\dfrac{r_{1} r_{2}}{\left(r_{1}+r_{2}\right) / 2}=\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\end{aligned}\)
PHXI08:GRAVITATION
359962
Average distance of the earth from the sun is \(L_{1}\). If one year of earth \(=D\) days, one year of another planet whose average distance from the sum is \(L_{2}\) will be
359958
A planet moving along an elliptical orbit is close to the sun at a distance \(r_{1}\) and farthest away at a distance of \(r_{2}\). If \(v_{1}\) and \(v_{2}\) are the linear velocities at these points respectively. Then the ratio \(\frac{{{v_1}}}{{{v_2}}}\) is
1 \(\dfrac{r_{2}}{r_{1}}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
Explanation:
From conservation of angular momentum of the planet of mass \(m\). \(\frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}\) [angular momentum is constant]
PHXI08:GRAVITATION
359959
The ratio of \(K E\) of a planet at the points 1 and 2 is
1 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\dfrac{r_{2}}{r_{1}}\)
Explanation:
\(v_{1} r_{1}=v_{2} r_{2} \Rightarrow \dfrac{v_{1}}{v_{2}}=\dfrac{r_{2}}{r_{1}}\) So, \(\dfrac{K_{1}}{K_{2}}=\left(\dfrac{v_{1}}{v_{2}}\right)^{2}=\dfrac{r_{2}^{2}}{r_{1}^{2}}\).
PHXI08:GRAVITATION
359960
If ' \(A\) ' is areal velocity of a planet of mass \(M\), its angular momentum is
1 \(M / A\)
2 \(2 M A\)
3 \(A^{2} M\)
4 \(A M^{2}\)
Explanation:
\(\dfrac{d A}{d t} \rightarrow\) Areal velocity of a planet revolving around sun. \(\dfrac{d A}{d t}=\dfrac{L}{2 M}, L \rightarrow\) Angular momentum of planet. \( \Rightarrow L = 2M\frac{{dA}}{{dt}} = 2\,MA\).
PHXI08:GRAVITATION
359961
The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
1 \(\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\)
2 \(\dfrac{r_{1}+r_{2}}{3}\)
3 \(\dfrac{r_{1}+r_{2}}{4}\)
4 \(\dfrac{r_{1} r_{2}}{r_{1}+r_{2}}\)
Explanation:
The earth moves around the sun in elliptical path as shown in the figure. So by using the properties of ellipse \(r_{1}=(1+e) a \text { and } r_{2}=(1-e) a\) where, \(a=\) length of semi-major axis \(b=\) length of semi-major axis \(e=\) eccentricity Given that \(\Rightarrow a=\dfrac{r_{1}+r_{2}}{2}\) and \(r_{1} r_{2}=\left(1-e^{2}\right) a^{2}\) Now, required distance \(=\) semi latusrectum \(\begin{aligned}F_{1} P & =\dfrac{b^{2}}{a} \\& =\dfrac{a^{2}\left(1-e^{2}\right)}{a}=\dfrac{r_{1} r_{2}}{\left(r_{1}+r_{2}\right) / 2}=\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\end{aligned}\)
PHXI08:GRAVITATION
359962
Average distance of the earth from the sun is \(L_{1}\). If one year of earth \(=D\) days, one year of another planet whose average distance from the sum is \(L_{2}\) will be
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PHXI08:GRAVITATION
359958
A planet moving along an elliptical orbit is close to the sun at a distance \(r_{1}\) and farthest away at a distance of \(r_{2}\). If \(v_{1}\) and \(v_{2}\) are the linear velocities at these points respectively. Then the ratio \(\frac{{{v_1}}}{{{v_2}}}\) is
1 \(\dfrac{r_{2}}{r_{1}}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
Explanation:
From conservation of angular momentum of the planet of mass \(m\). \(\frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}\) [angular momentum is constant]
PHXI08:GRAVITATION
359959
The ratio of \(K E\) of a planet at the points 1 and 2 is
1 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\dfrac{r_{2}}{r_{1}}\)
Explanation:
\(v_{1} r_{1}=v_{2} r_{2} \Rightarrow \dfrac{v_{1}}{v_{2}}=\dfrac{r_{2}}{r_{1}}\) So, \(\dfrac{K_{1}}{K_{2}}=\left(\dfrac{v_{1}}{v_{2}}\right)^{2}=\dfrac{r_{2}^{2}}{r_{1}^{2}}\).
PHXI08:GRAVITATION
359960
If ' \(A\) ' is areal velocity of a planet of mass \(M\), its angular momentum is
1 \(M / A\)
2 \(2 M A\)
3 \(A^{2} M\)
4 \(A M^{2}\)
Explanation:
\(\dfrac{d A}{d t} \rightarrow\) Areal velocity of a planet revolving around sun. \(\dfrac{d A}{d t}=\dfrac{L}{2 M}, L \rightarrow\) Angular momentum of planet. \( \Rightarrow L = 2M\frac{{dA}}{{dt}} = 2\,MA\).
PHXI08:GRAVITATION
359961
The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
1 \(\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\)
2 \(\dfrac{r_{1}+r_{2}}{3}\)
3 \(\dfrac{r_{1}+r_{2}}{4}\)
4 \(\dfrac{r_{1} r_{2}}{r_{1}+r_{2}}\)
Explanation:
The earth moves around the sun in elliptical path as shown in the figure. So by using the properties of ellipse \(r_{1}=(1+e) a \text { and } r_{2}=(1-e) a\) where, \(a=\) length of semi-major axis \(b=\) length of semi-major axis \(e=\) eccentricity Given that \(\Rightarrow a=\dfrac{r_{1}+r_{2}}{2}\) and \(r_{1} r_{2}=\left(1-e^{2}\right) a^{2}\) Now, required distance \(=\) semi latusrectum \(\begin{aligned}F_{1} P & =\dfrac{b^{2}}{a} \\& =\dfrac{a^{2}\left(1-e^{2}\right)}{a}=\dfrac{r_{1} r_{2}}{\left(r_{1}+r_{2}\right) / 2}=\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\end{aligned}\)
PHXI08:GRAVITATION
359962
Average distance of the earth from the sun is \(L_{1}\). If one year of earth \(=D\) days, one year of another planet whose average distance from the sum is \(L_{2}\) will be
359958
A planet moving along an elliptical orbit is close to the sun at a distance \(r_{1}\) and farthest away at a distance of \(r_{2}\). If \(v_{1}\) and \(v_{2}\) are the linear velocities at these points respectively. Then the ratio \(\frac{{{v_1}}}{{{v_2}}}\) is
1 \(\dfrac{r_{2}}{r_{1}}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
Explanation:
From conservation of angular momentum of the planet of mass \(m\). \(\frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}\) [angular momentum is constant]
PHXI08:GRAVITATION
359959
The ratio of \(K E\) of a planet at the points 1 and 2 is
1 \(\left(\dfrac{r_{1}}{r_{2}}\right)^{2}\)
2 \(\left(\dfrac{r_{2}}{r_{1}}\right)^{2}\)
3 \(\dfrac{r_{1}}{r_{2}}\)
4 \(\dfrac{r_{2}}{r_{1}}\)
Explanation:
\(v_{1} r_{1}=v_{2} r_{2} \Rightarrow \dfrac{v_{1}}{v_{2}}=\dfrac{r_{2}}{r_{1}}\) So, \(\dfrac{K_{1}}{K_{2}}=\left(\dfrac{v_{1}}{v_{2}}\right)^{2}=\dfrac{r_{2}^{2}}{r_{1}^{2}}\).
PHXI08:GRAVITATION
359960
If ' \(A\) ' is areal velocity of a planet of mass \(M\), its angular momentum is
1 \(M / A\)
2 \(2 M A\)
3 \(A^{2} M\)
4 \(A M^{2}\)
Explanation:
\(\dfrac{d A}{d t} \rightarrow\) Areal velocity of a planet revolving around sun. \(\dfrac{d A}{d t}=\dfrac{L}{2 M}, L \rightarrow\) Angular momentum of planet. \( \Rightarrow L = 2M\frac{{dA}}{{dt}} = 2\,MA\).
PHXI08:GRAVITATION
359961
The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is
1 \(\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\)
2 \(\dfrac{r_{1}+r_{2}}{3}\)
3 \(\dfrac{r_{1}+r_{2}}{4}\)
4 \(\dfrac{r_{1} r_{2}}{r_{1}+r_{2}}\)
Explanation:
The earth moves around the sun in elliptical path as shown in the figure. So by using the properties of ellipse \(r_{1}=(1+e) a \text { and } r_{2}=(1-e) a\) where, \(a=\) length of semi-major axis \(b=\) length of semi-major axis \(e=\) eccentricity Given that \(\Rightarrow a=\dfrac{r_{1}+r_{2}}{2}\) and \(r_{1} r_{2}=\left(1-e^{2}\right) a^{2}\) Now, required distance \(=\) semi latusrectum \(\begin{aligned}F_{1} P & =\dfrac{b^{2}}{a} \\& =\dfrac{a^{2}\left(1-e^{2}\right)}{a}=\dfrac{r_{1} r_{2}}{\left(r_{1}+r_{2}\right) / 2}=\dfrac{2 r_{1} r_{2}}{r_{1}+r_{2}}\end{aligned}\)
PHXI08:GRAVITATION
359962
Average distance of the earth from the sun is \(L_{1}\). If one year of earth \(=D\) days, one year of another planet whose average distance from the sum is \(L_{2}\) will be
