A rocket is moving away from the solar system at a speed of6.0  103 m/s. It fires its engine, which ejects exhaust with a speedof 3.0  103 m/s relative to the rocket. The mass of the rocket atthis time is 4.0  104 kg, and its acceleration is 2.0 m/s2. (a) What isthe thrust of the engine? (b) At what rate, in kilograms per second,is exhaust ejected during the firing?

To tackle the problem of the rocket's thrust and the rate of exhaust ejection, we can apply some fundamental principles from physics, particularly Newton's second law and the concept of momentum. Let's break it down step by step.

Calculating the Thrust of the Engine

The thrust produced by the rocket engine can be determined using Newton's second law, which states that force equals mass times acceleration (F = ma). In this case, the thrust is the net force acting on the rocket.

Given:

• Mass of the rocket (m) = 4.0 × 10⁴ kg
• Acceleration (a) = 2.0 m/s²

Using the formula:

Thrust (T) = m × a

Substituting the values:

T = (4.0 × 10⁴ kg) × (2.0 m/s²) = 8.0 × 10⁴ N

So, the thrust of the engine is 80,000 N.

Determining the Rate of Exhaust Ejection

Next, we need to find out how quickly the exhaust is being ejected from the rocket. To do this, we can use the principle of conservation of momentum. The thrust can also be expressed in terms of the mass flow rate of the exhaust and its velocity relative to the rocket.

The formula for thrust in terms of exhaust velocity (v_e) and mass flow rate (ṁ) is:

T = ṁ × v_e

Here, the exhaust velocity relative to the rocket is given as 3.0 × 10³ m/s. Since the rocket is moving away from the solar system at 6.0 × 10³ m/s, the effective exhaust velocity relative to an outside observer is the speed of the rocket plus the exhaust speed:

v_e = 6.0 × 10³ m/s + 3.0 × 10³ m/s = 9.0 × 10³ m/s

Now, we can rearrange the thrust equation to solve for the mass flow rate (ṁ):

ṁ = T / v_e

Substituting the values we have:

ṁ = (8.0 × 10⁴ N) / (9.0 × 10³ m/s) ≈ 8.89 kg/s

Thus, the rate at which exhaust is ejected during the firing is approximately 8.89 kg/s.

Summary of Results

In summary, we found that:

• The thrust of the engine is 80,000 N.
• The rate of exhaust ejection is approximately 8.89 kg/s.

This analysis illustrates how the principles of physics can be applied to understand the dynamics of rocket propulsion. If you have any further questions or need clarification on any part of this, feel free to ask!