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.