Question:

The escape velocity of a projectile on the earths surface is $ 11.2\,\,km{{s}^{-1}} $ . A body is projected out with thrice this speed. The speed of the body far away from the earth will be:

Updated On: Jun 20, 2022
  • $ 22.4\,km{{s}^{-1}} $
  • $ 31.7\,km{{s}^{-1}} $
  • $ 33.6\,km{{s}^{-1}} $
  • none of these
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The Correct Option is B

Solution and Explanation

Key Idea : Applying conservation of energy. By law of conservation of energy $(U+K)_{\text {surface }}=(U+K)_{\infty}$
$\Rightarrow \quad-\frac{G M m}{R}+\frac{1}{2} m\left(3 v_{e}\right)^{2}=0+\frac{1}{2} m v^{2}$
$\Rightarrow -\frac{G M}{R}+\frac{9 v_{e}^{2}}{2}=\frac{1}{2} v^{2}$
Since, $\nu_{e}^{2}=\frac{2 G M}{R}$
$\therefore -\frac{v_{e}^{2}}{2},+\frac{9 v_{e}^{2}}{2}=\frac{1}{2} v^{2}$
$\Rightarrow v^{2} =8 \,v_{e}^{2} $
$\therefore v =2 \sqrt{2} \,v_{e}$
$=2 \sqrt{2} \times 11.2$
$=31.7 \,kms ^{-1}$
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Concepts Used:

Motion in a straight line

The motion in a straight line is an object changes its position with respect to its surroundings with time, then it is called in motion. It is a change in the position of an object over time. It is nothing but linear motion. 

Types of Linear Motion:

Linear motion is also known as the Rectilinear Motion which are of two types:

  1. Uniform linear motion with constant velocity or zero acceleration: If a body travels in a straight line by covering an equal amount of distance in an equal interval of time then it is said to have uniform motion.
  2. Non-Uniform linear motion with variable velocity or non-zero acceleration: Not like the uniform acceleration, the body is said to have a non-uniform motion when the velocity of a body changes by unequal amounts in equal intervals of time. The rate of change of its velocity changes at different points of time during its movement.