A rocket is fired vertically with a speed of 5 `kms^(-1)` from the earth's surface. How far from the earth does the rocket go before returning to the earth ? Mass of earth=`6.0xx10^(24)`kg, mean radius of the earth =`6.4xx10^(6)`m, G=`6.67xx10^(-11)Nm^(2)Kg^(-2).`

Explanation

To solve the problem of how far a rocket goes before returning to Earth, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The rocket is fired vertically with an initial speed \( v = 5 \, \text{km/s} = 5000 \, \text{m/s} \) from the Earth's surface. We need to find the maximum height \( h \) it reaches before falling back to Earth.

### Step 2: Write the Conservation of Energy Equation
The total mechanical energy at the surface of the Earth (initial) is equal to the total mechanical energy at the maximum height (final).

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