Problem 53 - College Physics

Question

A small rocket burns 0.0500 \(\mathrm{kg}\) of fuel per second, ejecting it as a gas with a velocity of magnitude 1600 \(\mathrm{m} / \mathrm{s}\) relative to the rocket. (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

(a) The thrust is 80 N. (b) The rocket can operate in outer space. Steering is done by adjusting exhaust direction, and braking is possible by firing thrusters in the opposite direction.

Step 1: Identify Given Information

We are given that the rocket burns 0.0500 kg of fuel per second and ejects it at a velocity of 1600 m/s relative to the rocket.

Step 2: Determine the Thrust Formula

Thrust (T) is determined using the formula: \[ T = \dot{m} \cdot v_{e} \]where \( \dot{m} \) is the rate of change of mass (mass flow rate) and \( v_{e} \) is the exhaust velocity.

Step 3: Calculate the Thrust

Substitute \( \dot{m} = 0.0500 \ \mathrm{kg/s} \) and \( v_{e} = 1600 \ \mathrm{m/s} \) into the thrust formula:\[ T = 0.0500 \ \mathrm{kg/s} \cdot 1600 \ \mathrm{m/s} = 80 \ \mathrm{N} \].Thus, the thrust of the rocket is 80 N.

Step 4: Assess Operation in Outer Space

In outer space, there is no atmospheric resistance. The rocket operates on the principle of conservation of momentum, which does not require an atmosphere. Hence, the rocket will operate in outer space.

Step 5: Steering and Braking in Outer Space

The rocket can be steered by adjusting the direction of the exhaust gases, as altering the direction will change the thrust direction. To brake or decrease speed, the rocket would need to fire in the opposite direction of travel, using its thrusters.

Key Concepts

Thrust Calculation

Thrust is a crucial concept in rocket physics. It refers to the force exerted by the rocket engine to propel the vehicle forward. This force is generated by expelling mass (such as gas) at high speeds. The thrust calculation helps determine the force exerted by the rocket.

To find the thrust, we use the formula: \[ T = \dot{m} \cdot v_{e} \]where:

• \(T\) is the thrust force measured in Newtons (N),
• \(\dot{m}\) is the mass flow rate (the amount of fuel burnt per second), measured in kilograms per second (kg/s), and
• \(v_{e}\) is the exhaust velocity of the gases, measured in meters per second (m/s).

In our exercise, the rocket burns 0.0500 kg of fuel every second and expels it with a velocity of 1600 m/s. By plugging these values into the equation, we find that the thrust is 80 Newtons. This means the rocket exerts a force of 80 N to propel itself forward.

Conservation of Momentum

The concept of conservation of momentum is fundamental in understanding how rockets work, especially in the vacuum of space.

Momentum is the product of an object's mass and velocity. According to the law of conservation of momentum, the total momentum of a closed system remains constant if no external forces act on it. This principle applies perfectly in the vacuum of space where there is no atmospheric interference.

For rockets, this means that when they expel fuel in one direction, they move in the opposite direction. When the rocket expels gas with high velocity (as given in the exercise), it gains speed in the opposite direction to conserve the system's overall momentum. This is how rockets can operate and travel manoeuvre into the vast emptiness of space.

Exhaust Velocity

Exhaust velocity is the speed at which gas exits the rocket engine. It plays a vital role in determining the efficiency and effectiveness of the propulsion system.

In our exercise scenario, the exhaust velocity is 1600 m/s. This high speed helps the rocket to produce considerable thrust.

The relationship between thrust and exhaust velocity is direct. Higher exhaust velocity means greater thrust for the same fuel consumption rate. Achieving high exhaust velocity is key to efficient space travel—a prime reason why rocket engineers invest in innovative technologies to maximize this aspect of rocket design.

Outer Space Operation

Operating a rocket in outer space is vastly different from doing so within Earth's atmosphere. In space, there is no air resistance or gravity acting like it would on Earth. Rockets utilize the conservation of momentum to function, which makes them perfectly capable of operating in space.

Without an atmosphere, rockets rely solely on their own systems to maneuver. Steering a rocket is done by changing the direction of the thrust, which can be achieved through gimbaling the engines or using smaller thrusters for adjusting its path.

Braking in space is another interesting challenge. It is done by firing the rocket engines in the opposite direction to the travel path, slowing the vehicle down. However, this requires careful fuel planning, as every maneuver consumes fuel—an important consideration for spacecraft traveling long distances.