359818
The ratio of the radii of planets \(A\) and \(B\) is \(k_{1}\) and ratio of acceleration due to gravity on them is \(k_{2}\). The ratio of escape velocities from them will be
1 \(\sqrt{k_{1} k_{2}}\)
2 \(\sqrt{\dfrac{k_{1}}{k_{2}}}\)
3 \(k_{1} k_{2}\)
4 \(\sqrt{\dfrac{k_{2}}{k_{1}}}\)
Explanation:
The escape velocity is \(\begin{gathered}v=\sqrt{2 g R} \\\Rightarrow \dfrac{v_{A}}{v_{B}}=\sqrt{\dfrac{g_{A}}{g_{B}} \times \dfrac{R_{A}}{R_{B}}}=\sqrt{k_{1} \times k_{2}}=\sqrt{k_{1} k_{2}}\end{gathered}\)
PHXI08:GRAVITATION
359819
The velocity with which a projectile must be fired so that it escapes earth's gravitation does not depend on:
1 Mass of the earth
2 Mass of the projectile
3 Radius of the projectile's orbit
4 Gravitational constant
Explanation:
The escape velocity of a body is \(\therefore \quad v_{e}=\sqrt{\dfrac{2 G M_{e}}{R_{e}}}\) The above formula shows that escape velocity is independent of the mass of the projectile.
PHXI08:GRAVITATION
359820
Two planets \(A\) and \(B\) have the same material density. If the radius of \(A\) is twice that of \(B\), then the ratio of the escape velocity \(v_{A} / v_{B}\) is
1 2
2 \(\sqrt{2}\)
3 \(1 / \sqrt{2}\)
4 \(1 / 2\)
Explanation:
\({v_e} = \sqrt {2gR} = \sqrt {\frac{{2GM}}{{{R^2}}}R} = \sqrt {\frac{{2Gd\frac{4}{3}\pi {R^3}}}{{{R^2}}}} R\) \( = R\sqrt {2Gd\frac{4}{3}\pi } \) as \({v_e} \propto R\) for same density, \(\frac{{{v_A}}}{{{v_B}}} = 2\)
PHXI08:GRAVITATION
359821
Assertion : Earth has an atmosphere but the moon does not. Reason : Smaller escape velocity in the moon makes the moon to have no atmosphere.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The escape velocity of a gas on the surface of the Moon is relatively low, only \(2.5\;km{\rm{/}}s\).\(v_{e}=\sqrt{\dfrac{2 G M}{R}}\). \(M\) is smaller for moon as a result, all the gas molecules have escaped, leading to the absence of an atmosphere on the Moon. So correct option is (1).
359818
The ratio of the radii of planets \(A\) and \(B\) is \(k_{1}\) and ratio of acceleration due to gravity on them is \(k_{2}\). The ratio of escape velocities from them will be
1 \(\sqrt{k_{1} k_{2}}\)
2 \(\sqrt{\dfrac{k_{1}}{k_{2}}}\)
3 \(k_{1} k_{2}\)
4 \(\sqrt{\dfrac{k_{2}}{k_{1}}}\)
Explanation:
The escape velocity is \(\begin{gathered}v=\sqrt{2 g R} \\\Rightarrow \dfrac{v_{A}}{v_{B}}=\sqrt{\dfrac{g_{A}}{g_{B}} \times \dfrac{R_{A}}{R_{B}}}=\sqrt{k_{1} \times k_{2}}=\sqrt{k_{1} k_{2}}\end{gathered}\)
PHXI08:GRAVITATION
359819
The velocity with which a projectile must be fired so that it escapes earth's gravitation does not depend on:
1 Mass of the earth
2 Mass of the projectile
3 Radius of the projectile's orbit
4 Gravitational constant
Explanation:
The escape velocity of a body is \(\therefore \quad v_{e}=\sqrt{\dfrac{2 G M_{e}}{R_{e}}}\) The above formula shows that escape velocity is independent of the mass of the projectile.
PHXI08:GRAVITATION
359820
Two planets \(A\) and \(B\) have the same material density. If the radius of \(A\) is twice that of \(B\), then the ratio of the escape velocity \(v_{A} / v_{B}\) is
1 2
2 \(\sqrt{2}\)
3 \(1 / \sqrt{2}\)
4 \(1 / 2\)
Explanation:
\({v_e} = \sqrt {2gR} = \sqrt {\frac{{2GM}}{{{R^2}}}R} = \sqrt {\frac{{2Gd\frac{4}{3}\pi {R^3}}}{{{R^2}}}} R\) \( = R\sqrt {2Gd\frac{4}{3}\pi } \) as \({v_e} \propto R\) for same density, \(\frac{{{v_A}}}{{{v_B}}} = 2\)
PHXI08:GRAVITATION
359821
Assertion : Earth has an atmosphere but the moon does not. Reason : Smaller escape velocity in the moon makes the moon to have no atmosphere.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The escape velocity of a gas on the surface of the Moon is relatively low, only \(2.5\;km{\rm{/}}s\).\(v_{e}=\sqrt{\dfrac{2 G M}{R}}\). \(M\) is smaller for moon as a result, all the gas molecules have escaped, leading to the absence of an atmosphere on the Moon. So correct option is (1).
359818
The ratio of the radii of planets \(A\) and \(B\) is \(k_{1}\) and ratio of acceleration due to gravity on them is \(k_{2}\). The ratio of escape velocities from them will be
1 \(\sqrt{k_{1} k_{2}}\)
2 \(\sqrt{\dfrac{k_{1}}{k_{2}}}\)
3 \(k_{1} k_{2}\)
4 \(\sqrt{\dfrac{k_{2}}{k_{1}}}\)
Explanation:
The escape velocity is \(\begin{gathered}v=\sqrt{2 g R} \\\Rightarrow \dfrac{v_{A}}{v_{B}}=\sqrt{\dfrac{g_{A}}{g_{B}} \times \dfrac{R_{A}}{R_{B}}}=\sqrt{k_{1} \times k_{2}}=\sqrt{k_{1} k_{2}}\end{gathered}\)
PHXI08:GRAVITATION
359819
The velocity with which a projectile must be fired so that it escapes earth's gravitation does not depend on:
1 Mass of the earth
2 Mass of the projectile
3 Radius of the projectile's orbit
4 Gravitational constant
Explanation:
The escape velocity of a body is \(\therefore \quad v_{e}=\sqrt{\dfrac{2 G M_{e}}{R_{e}}}\) The above formula shows that escape velocity is independent of the mass of the projectile.
PHXI08:GRAVITATION
359820
Two planets \(A\) and \(B\) have the same material density. If the radius of \(A\) is twice that of \(B\), then the ratio of the escape velocity \(v_{A} / v_{B}\) is
1 2
2 \(\sqrt{2}\)
3 \(1 / \sqrt{2}\)
4 \(1 / 2\)
Explanation:
\({v_e} = \sqrt {2gR} = \sqrt {\frac{{2GM}}{{{R^2}}}R} = \sqrt {\frac{{2Gd\frac{4}{3}\pi {R^3}}}{{{R^2}}}} R\) \( = R\sqrt {2Gd\frac{4}{3}\pi } \) as \({v_e} \propto R\) for same density, \(\frac{{{v_A}}}{{{v_B}}} = 2\)
PHXI08:GRAVITATION
359821
Assertion : Earth has an atmosphere but the moon does not. Reason : Smaller escape velocity in the moon makes the moon to have no atmosphere.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The escape velocity of a gas on the surface of the Moon is relatively low, only \(2.5\;km{\rm{/}}s\).\(v_{e}=\sqrt{\dfrac{2 G M}{R}}\). \(M\) is smaller for moon as a result, all the gas molecules have escaped, leading to the absence of an atmosphere on the Moon. So correct option is (1).
359818
The ratio of the radii of planets \(A\) and \(B\) is \(k_{1}\) and ratio of acceleration due to gravity on them is \(k_{2}\). The ratio of escape velocities from them will be
1 \(\sqrt{k_{1} k_{2}}\)
2 \(\sqrt{\dfrac{k_{1}}{k_{2}}}\)
3 \(k_{1} k_{2}\)
4 \(\sqrt{\dfrac{k_{2}}{k_{1}}}\)
Explanation:
The escape velocity is \(\begin{gathered}v=\sqrt{2 g R} \\\Rightarrow \dfrac{v_{A}}{v_{B}}=\sqrt{\dfrac{g_{A}}{g_{B}} \times \dfrac{R_{A}}{R_{B}}}=\sqrt{k_{1} \times k_{2}}=\sqrt{k_{1} k_{2}}\end{gathered}\)
PHXI08:GRAVITATION
359819
The velocity with which a projectile must be fired so that it escapes earth's gravitation does not depend on:
1 Mass of the earth
2 Mass of the projectile
3 Radius of the projectile's orbit
4 Gravitational constant
Explanation:
The escape velocity of a body is \(\therefore \quad v_{e}=\sqrt{\dfrac{2 G M_{e}}{R_{e}}}\) The above formula shows that escape velocity is independent of the mass of the projectile.
PHXI08:GRAVITATION
359820
Two planets \(A\) and \(B\) have the same material density. If the radius of \(A\) is twice that of \(B\), then the ratio of the escape velocity \(v_{A} / v_{B}\) is
1 2
2 \(\sqrt{2}\)
3 \(1 / \sqrt{2}\)
4 \(1 / 2\)
Explanation:
\({v_e} = \sqrt {2gR} = \sqrt {\frac{{2GM}}{{{R^2}}}R} = \sqrt {\frac{{2Gd\frac{4}{3}\pi {R^3}}}{{{R^2}}}} R\) \( = R\sqrt {2Gd\frac{4}{3}\pi } \) as \({v_e} \propto R\) for same density, \(\frac{{{v_A}}}{{{v_B}}} = 2\)
PHXI08:GRAVITATION
359821
Assertion : Earth has an atmosphere but the moon does not. Reason : Smaller escape velocity in the moon makes the moon to have no atmosphere.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The escape velocity of a gas on the surface of the Moon is relatively low, only \(2.5\;km{\rm{/}}s\).\(v_{e}=\sqrt{\dfrac{2 G M}{R}}\). \(M\) is smaller for moon as a result, all the gas molecules have escaped, leading to the absence of an atmosphere on the Moon. So correct option is (1).
