A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at 2500 m/s . It is loaded with 600 kg of fuel, which it burns in 30 s. What is the rocket's speed 10 s, 20 s, and 30 s after launch?

Verified step by step guidance

1. Step 1: Use the rocket equation, which is derived from the conservation of momentum. The equation is \( v = v_e \ln\left(\frac{m_i}{m_f}\right) \), where \( v \) is the final velocity of the rocket, \( v_e \) is the exhaust velocity of the gases, \( m_i \) is the initial total mass (rocket plus fuel), and \( m_f \) is the final mass (rocket only).
2. Step 2: Calculate the initial mass \( m_i \) of the rocket at launch by adding the mass of the rocket and the mass of the fuel: \( m_i = 150 \, \text{kg} + 600 \, \text{kg} = 750 \, \text{kg} \).
3. Step 3: Determine the mass of the rocket at different times as the fuel burns. The fuel burn rate is \( \frac{600 \, \text{kg}}{30 \, \text{s}} = 20 \, \text{kg/s} \). At 10 s, 20 s, and 30 s, calculate the remaining fuel and the total mass at each time point.
4. Step 4: Apply the rocket equation at each time point to find the speed. For example, at 10 s, the remaining fuel is \( 600 \, \text{kg} - 20 \, \text{kg/s} \times 10 \, \text{s} = 400 \, \text{kg} \), so the total mass is \( 150 \, \text{kg} + 400 \, \text{kg} = 550 \, \text{kg} \). Plug these values into the rocket equation to find the speed at 10 s.
5. Step 5: Repeat the calculation for 20 s and 30 s using the respective remaining fuel and total mass to find the speeds at those times.

Key Concepts

Conservation of Momentum

The conservation of momentum principle states that in a closed system, the total momentum remains constant if no external forces act on it. In the context of a rocket, as it expels gas, the momentum gained by the rocket is equal and opposite to the momentum lost by the expelled gas, allowing us to calculate the rocket's speed at various times.

Rocket Equation (Tsiolkovsky's Equation)

Tsiolkovsky's rocket equation relates the velocity of a rocket to its mass and the velocity of the exhaust gases. It is expressed as Δv = ve * ln(m0/mf), where Δv is the change in velocity, ve is the effective exhaust velocity, m0 is the initial mass, and mf is the final mass. This equation is crucial for determining the speed of the rocket as it burns fuel.

Thrust and Fuel Consumption

Thrust is the force that propels the rocket forward, generated by the expulsion of exhaust gases. The rate of fuel consumption affects the rocket's mass over time, which in turn influences its acceleration and speed. Understanding how thrust and fuel consumption interact is essential for calculating the rocket's speed at different intervals after launch.