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: Understanding the Concept
The minimum kinetic energy to escape from the surface is \(\frac{GMm}{R}\). Here the body is given half of it, \(\frac{GMm}{2R}\).

Step 2: Energy at the surface and at the top
Initial energy: \(\frac{GMm}{2R}-\frac{GMm}{R}=-\frac{GMm}{2R}\).
At the maximum height \(h\) the speed is zero: \(-\frac{GMm}{R+h}\).

Step 3: Calculation
Equate the two:
\(\frac{1}{R+h}=\frac{1}{2R}\Rightarrow R+h=2R\)
So \(h=R\).

Step 4: Compare the options
The height \(R\) is option (C).

Final Answer:
The body rises to a height equal to R. \[ \boxed{\text{(C) }R} \]