An object is kept at rest at a distance of 3R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant)

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Q.
An object is kept at rest at a distance of 3R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant) [2025]

1 $\sqrt{\frac{G M}{2 R}}$

2 $\sqrt{\frac{G M}{R}}$

3 $\sqrt{\frac{3 G M}{R}}$

4 $\sqrt{\frac{2 G M}{R}}$

Ans.
(1)

$K E_{i} + P E_{i} = K E_{g} + P E_{g}$

$\frac{1}{2} m v^{2} - \frac{G M m}{4 R} = 0 + 0$

$v^{2} = \frac{G M}{R} \frac{1}{2} \Rightarrow v = \sqrt{\frac{G M}{2 R}}$

Solution

Q.
An object is kept at rest at a distance of 3R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant) [2025]

1 $\sqrt{\frac{G M}{2 R}}$

2 $\sqrt{\frac{G M}{R}}$

3 $\sqrt{\frac{3 G M}{R}}$

4 $\sqrt{\frac{2 G M}{R}}$

Ans.
(1)

$K E_{i} + P E_{i} = K E_{g} + P E_{g}$

$\frac{1}{2} m v^{2} - \frac{G M m}{4 R} = 0 + 0$

$v^{2} = \frac{G M}{R} \frac{1}{2} \Rightarrow v = \sqrt{\frac{G M}{2 R}}$

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Solution

Q. An object is kept at rest at a distance of 3R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant) [2025]

1 GM2R

2 GMR

3 3GMR

4 2GMR

Ans. (1) KEi+PEi=KEg+PEg 12mv2–GMm4R=0+0 v2=GMR12 ⇒ v=GM2R

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Q.
An object is kept at rest at a distance of 3R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant) [2025]

1 $\sqrt{\frac{G M}{2 R}}$

2 $\sqrt{\frac{G M}{R}}$

3 $\sqrt{\frac{3 G M}{R}}$

4 $\sqrt{\frac{2 G M}{R}}$

Ans.
(1)

$K E_{i} + P E_{i} = K E_{g} + P E_{g}$

$\frac{1}{2} m v^{2} - \frac{G M m}{4 R} = 0 + 0$

$v^{2} = \frac{G M}{R} \frac{1}{2} \Rightarrow v = \sqrt{\frac{G M}{2 R}}$