A rocket is fired fired vertically from the surface of Mars with a speed of 2 `Kms^(-1)`. If `20%` of its initial energy is lost due to martain atmospheric resistance, how far will the rocket go from the surface of Mars before returning to it ? Mass of Mars=`6.4xx10^(23)`Kg, radius of Mars=3395 Km, G=`6.67xx10^(-11)Nm^(2)Kg^(-2)`.

Explanation

To solve the problem step by step, we will use the principles of energy conservation and the equations of motion under gravity.

### Step 1: Understand the Initial Conditions
The rocket is fired vertically with an initial speed \( v_0 = 2 \, \text{km/s} = 2000 \, \text{m/s} \). We also know that 20% of its initial energy is lost due to atmospheric resistance, meaning that only 80% of the initial kinetic energy will be used to reach the maximum height.

### Step 2: Calculate the Initial Kinetic Energy
The initial kinetic energy (KE) of the rocket can be calculated using the formula:
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