Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Neglecting air resistance and the rotation of the earth, the minimum speed with which a body must be projected from the earth's surface to escape is the same whether it is projected vertically upward or at $45^\circ$ to the vertical.
Reason (R): The gravitational force is conservative, so whether a body escapes depends only on its total mechanical energy at launch, not on the direction of its velocity.
In the light of the above statements, choose the correct answer from the options given below:
Answer: (A) Both (A) and (R) are true and (R) is the correct explanation of (A)
Concept. A body escapes if its total mechanical energy is at least zero. Kinetic energy depends on speed, not direction, so the condition fixes a speed only.
Step 1. Assertion. At launch
$$\frac12mv^2 - \frac{GMm}{R} \ge 0 \;\Rightarrow\; v \ge \sqrt{\frac{2GM}{R}}$$
The direction does not appear, so the escape speed is the same at any angle (unless the path hits the earth, which does not happen for an upward launch). A is true.
Step 2. Reason. The work done by gravity depends only on the end points because the force is conservative. So energy conservation applies and only the launch energy matters. R is true, and it is exactly why A holds.
Answer. Both (A) and (R) are true and (R) is the correct explanation of (A) (Option A).
Solution by Sreeraj P, M.Sc Physics