Problem 78 - A Complete Resource Book in Physics for JEE Main

Question

The mass of a spaceship is \(1000 \mathrm{~kg}\). It is to be launched from the earth's surface out into free space. The value of \(g\) and \(r\) (radius of earth) are \(10 \mathrm{~m} / \mathrm{s}^{2}\) and \(6400 \mathrm{~km}\) respectively. The required energy for this work will be (A) \(6.4 \times 10^{11} \mathrm{~J}\) (B) \(6.4 \times 10^{8} \mathrm{~J}\) (C) \(6.4 \times 10^{9} \mathrm{~J}\) (D) \(6.4 \times 10^{10} \mathrm{~J}\)

Step-by-Step Solution

Answer

The short answer is: The required energy for this work is \(6.4 \times 10^{11} \mathrm{~J}\) (Option A).

Step 1: Identify the known values

We know the following values: Mass of spaceship (m) = 1000 kg Gravitational acceleration (g) = 10 m/s² Radius of Earth (r) = 6400 km = 6400 × 10^3 m

Step 2: Calculate the gravitational potential energy on Earth's surface

We will use the formula for gravitational potential energy (PE): PE = m × g × h In this case, h (height) is equal to the radius of the Earth. Therefore, we have: PE = 1000 kg × 10 m/s² × 6400 × 10^3 m PE = 6.4 × 10^7 kg × m²/s² or Joules (J)

Step 3: Calculate the energy required to move the spaceship to infinity

We will use the formula for gravitational potential energy at infinity: PE_infinity = - G × (m × M) / r Where G is the gravitational constant (approximately 6.674 × 10^-11 Nm²/kg²), M is the mass of Earth (approximately 5.972 × 10^24 kg), and r is the radius of Earth (6400 × 10^3 m). PE_infinity = - (6.674 × 10^-11 Nm²/kg²) × (1000 kg × 5.972 × 10^24 kg) / (6400 × 10^3 m) PE_infinity ≈ - 6.3934 × 10^11 J

Step 3: Calculate the total energy required to launch the spaceship

The total energy required to launch the spaceship is the difference between the gravitational potential energy on Earth's surface and the energy needed to move it to infinity: Energy_required = PE_infinity - PE Energy_required ≈ - 6.3934 × 10^11 J - 6.4 × 10^7 J Energy_required ≈ 6.4 × 10^11 J Comparing this energy value with the given options, we find that the answer is: (A) 6.4 × 10^11 J