Question

A rocket and its payload have a total mass of \(51,000 \mathrm{~kg} .\) How large is the thrust of the rocket engine when \((a)\) the rocket is "hovering" over the launch pad, just after ignition, and (b) when the rocket is accelerating upward at \(18 \mathrm{~m} / \mathrm{s}^{2} ?\)

Short answer

The thrust of the rocket is \(499,800\,N\) when the rocket is hovering, and \(1,417,800\,N\) when the rocket is accelerating upwards at \(18\,m/s^2\).

Step-by-step solution

Calculate the weight of the rocket

Weight is a force exerted by gravity on the object. We calculate it using the formula \(F = mg\), where \(m = 51,000\,kg\) is the mass of the rocket and \(g = 9.8\,m/s^2\) is the acceleration due to gravity. So the weight \(F_g = 51,000\,kg \times 9.8\,m/s^2 = 499,800\,N\). This gives us the force required to just hold the rocket stationary against the force of gravity.

Calculate the thrust when the rocket is hovering

When the rocket is hovering, it is not moving, and hence not accelerating. So according to Newton's second law, the net force acting on the rocket is zero. Hence, the thrust of the rocket is equal to the weight of the rocket. Therefore, the thrust \(F_t = F_g = 499,800\,N\).

Calculate the thrust when the rocket is accelerating upward

When the rocket is accelerating upward, it is experiencing two forces - the weight downward and the thrust upward. According to Newton's second law, the net force is the mass times the acceleration, so \(F = ma\). This net force is also the difference between the thrust and the weight (the thrust minus the weight). Therefore, \(ma = F_t - F_g\). From this we can find the thrust when the rocket is accelerating upward: \(F_t = ma + F_g = 51,000\,kg \times 18\,m/s^2 + 499,800\,N = 1,417,800\,N\).

Key concepts

Newton's Second Law of Motion

Newton's second law of motion is essential when understanding how forces affect the motion of objects. It can be succinctly stated as, \( F = ma \), which means that the force \( F \) acting upon an object is equal to the mass \( m \) of the object multiplied by the acceleration \( a \) that the object experiences.

This law explains how the amount of force needed to move an object increases with the object's mass and how the same force will cause a higher acceleration on an object with a smaller mass. In the context of a rocket launch, the thrust produced by the rocket's engine must overcome the gravitational force acting on the rocket to initiate movement. Thus, as the rocket mass or acceleration requirement increases, so does the demanded thrust.

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, it gives weight to physical objects and causes the downward acceleration of free-falling objects. The gravitational force acting on an object is calculated through the formula \( F_g = mg \), where \( F_g \) is the gravitational force, \( m \) is the mass, and \( g \) stands for the acceleration due to gravity.

Even rockets when on the launch pad are subject to Earth's gravitational force which pulls them towards the ground. The magnitude of this force is directly proportional to the mass of the rocket, a concept that significantly influences rocket thrust calculations.

Acceleration Due to Gravity

The acceleration due to gravity, denoted as \( g \), is the acceleration gained by an object due to the gravitational force. On the surface of the Earth, this value is approximately \( 9.8\text{ m/s}^2 \), which means that in the absence of air resistance, an object will increase its velocity by \( 9.8\text{ m/s} \) every second when it is in free fall.

In the context of a hovering rocket, the acceleration due to gravity plays a critical role as it defines the strength of the gravitational force the rocket must overcome to maintain its position or to launch. On planets with different gravitational strengths, the acceleration due to gravity changes, subsequently affecting the thrust required for rockets to leave the planet's surface.

Mass

Mass is a fundamental property of physical objects and is a quantitative measure of an object's resistance to acceleration when a force is applied. It is also a measure of the object's inertia, the tendency of an object to remain at constant velocity unless acted upon by a force. In the SI system, mass is measured in kilograms (kg).

The mass of the rocket, including its payload, is crucial in rocket thrust calculations, as directly indicated by Newton's second law of motion. The greater the mass of the rocket, the more thrust is needed to accelerate it. Understanding the relationship between a rocket's mass, the gravitational force it experiences, and the required thrust to achieve a certain acceleration is key to designing efficient and successful rocket launches.