The mass of a spaceship is 1000kg. It is to be launched from the earth's surface out into free space. The value of g and R (radius of earth) are `10m/s^2` and 6400 km respectively. The required energy for this work will be:

Options

A. `6.4xx10^(11) Joules`
B. `6.4xx10^8 Joules`
C. `6.4xx10^9 Joules`
D. `6.4xx10^(10) Joules`

Explanation

To calculate the required energy for launching the spaceship from the Earth's surface into free space, we can use the formula for gravitational potential energy. The energy required to move an object from the surface of the Earth to a point far away (essentially at infinity) can be calculated using the formula:

\[ E = \frac{G \cdot M \cdot m}{R} \]

Where:
- \(E\) is the energy required,
- \(G\) is the gravitational constant,
- \(M\) is the mass of the Earth,
- \(m\) is the mass of the spaceship,
- \(R\) is the radius of the Earth.

However, since we are given the value of \(g\) (acceleration due to gravity) at the Earth's surface, we can simplify our calculations. We know that:

\[ g = \frac{G \cdot M}{R^2} \]

From this, we can express \(G \cdot M\) as:

\[ G \cdot M = g \cdot R^2 \]

Now substituting this into our energy formula:

\[ E = \frac{g \cdot R^2 \cdot m}{R} = g \cdot R \cdot m \]

Now, we can plug in the values given in the problem:

- \(g = 10 \, \text{m/s}^2\)
- \(R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m}\)
- \(m = 1000 \, \text{kg}\)

Now substituting these values into the equation:

\[ E = 10 \, \text{m/s}^2 \cdot (6.4 \times 10^6 \, \text{m}) \cdot (1000 \, \text{kg}) \]

Calculating this gives:

\[ E = 10 \cdot 6.4 \times 10^6 \cdot 1000 \]
\[ E = 10 \cdot 6.4 \cdot 10^9 \]
\[ E = 64 \times 10^9 \, \text{J} \]
\[ E = 6.4 \times 10^{10} \, \text{J} \]

Thus, the required energy for launching the spaceship into free space is:

\[ \boxed{6.4 \times 10^{10} \, \text{J}} \]