Problem 62 - University Physics with Modern Physics

Question

A small rocket burns 0.0500 \(\mathrm{kg}\) of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 \(\mathrm{m} / \mathrm{s}\) . (a) What is the thrust of the rocket? (b) Would the rocket operate in outer space where there is no atmosphere? If so, how would you steer it? Could you brake it?

Step-by-Step Solution

Answer

The thrust of the rocket is 80 N. It can operate in space and be steered with thrusters, braking using retrorockets.

Step 1: Understanding Thrust

The thrust of a rocket is the force exerted by the propelling gas being ejected from its engine. It is calculated using the formula: \[ T = \frac{dm}{dt} \times v \] where \( T \) is the thrust, \( \frac{dm}{dt} \) is the rate of change of mass (mass flow rate of fuel), and \( v \) is the velocity of the ejected gas relative to the rocket.

Step 2: Calculate Mass Flow Rate

Given that the rocket burns 0.0500 kg of fuel per second, the rate of change of mass (mass flow rate) \( \frac{dm}{dt} \) is 0.0500 kg/s.

Step 3: Calculate Thrust

Substitute \( \frac{dm}{dt} = 0.0500 \) kg/s and \( v = 1600 \) m/s into the thrust formula:\[ T = 0.0500 \times 1600 = 80 \text{ N} \]The thrust of the rocket is 80 N.

Step 4: Rocket Operation in Space

A rocket can operate in outer space despite the absence of an atmosphere because it relies on the ejection of gas for propulsion, not the surrounding environment. Thus, the principles of conservation of momentum, not atmospheric pressure, allow it to operate.

Step 5: Steering and Braking in Space

A rocket can be steered in outer space by adjusting the direction of the thrusters or using gimbaled engines, which change the direction of the thrust vector. Braking can be achieved by reversing the direction of thrust or using additional retrorockets.

Key Concepts

Thrust Calculation

When launching a rocket, understanding thrust is essential. Thrust is the force exerted by the expelled gas from the rocket engine. The formula to calculate thrust is:\[ T = \frac{dm}{dt} \times v \]Where:

• \( T \) represents the thrust, measured in Newtons (N).
• \( \frac{dm}{dt} \) is the mass flow rate, or how much fuel the rocket burns per second, measured in kg/s.
• \( v \) stands for the speed at which the gas leaves the rocket, measured in m/s.

To find out the thrust of a specific rocket, you insert the known variables:

• Fuel burn rate: 0.0500 kg/s
• Gas ejection speed: 1600 m/s

Plugging these into the formula, we see:\[ T = 0.0500 \times 1600 = 80 \text{ N} \]Thus, the rocket generates a thrust of 80 Newtons, indicating the force it propels forward.

Operation in Space

The concept of rocket operation in space might seem puzzling without an atmosphere, but it's entirely possible. Imagine a balloon flying around when you release its air; it moves due to the expulsion of air, similar to how a rocket operates through expelling gas. Rocket propulsion doesn't rely on the surrounding atmosphere but rather on Newton's Third Law of Motion, which is about action and reaction. This principle dictates that every action has an equal and opposite reaction. Here's why rockets work in space:

• Rocket engines push gas out explosively, creating a pushing force back onto the rocket.
• Despite the absence of air, this propulsion mechanism remains effective.
• As the fuel burns, the ejected gases push the rocket forward. This is independent of atmospheric conditions.

Therefore, rockets operate effectively and efficiently in the vacuum of space.

Steering and Braking in Space

Steering and braking in space are fascinating parts of rocket science, not as straightforward as on Earth. Since rockets can't rely on wheels or friction, they must use thrust to maneuver. ### Steering in Space Steering is managed by altering the direction of the thrust. Rockets are equipped with:

• Gimbaled engines: engines on pivots to direct thrust in different directions.
• Thrusters placed strategically around the craft to fine-tune control and adjust orientation.

Through these methods, spacecraft can steer and change direction carefully. ### Braking in Space Braking requires reversing thrust to slow down. This can be done by:

• Firing retrorockets, smaller engines that direct thrust in the opposite direction.
• Adjusting main engines to counteract forward motion.

Both steering and braking involve precise control over thrust direction and magnitude, ensuring the spacecraft moves as intended in the vastness of space.