Question

A body is projected vertically upwards from earth's surface. If its kinetic energy of projection is equal to half of its minimum value required to escape from the gravitational influence then the height up to which it rises is (\(R\) = radius of earth)

• A. \(4R\)
• B. \(3R\)
• C. \(R\)
• D. \(2R\)

Hint:

Apply conservation of mechanical energy using the minimum escape kinetic energy GMm/R and set the speed to zero at the top.

Answer 1

The Correct Option is C

Step 1: Use the escape speed
The kinetic energy $\frac{1}{2}mv^{2}=\frac{1}{2}\cdot\frac{GMm}{R}$ means $v^{2}=\frac{GM}{R}$.

Step 2: Use the height formula
For projection speed $v$, the height satisfies $\frac{1}{2}mv^{2}=GMm\left(\frac{1}{R}-\frac{1}{R+h}\right)$.

Step 3: Solve
$\frac{1}{2R}=\frac{1}{R}-\frac{1}{R+h}\Rightarrow\frac{1}{R+h}=\frac{1}{2R}$, so $h=R$, option (C).

Final Answer:
The maximum height reached is R. \[ \boxed{\text{(C) }R} \]