You're an engineer working on Ares I, NASA's replacement for the space shuttles. Performance specs call for a first-stage rocket capable of accelerating a total mass of \(630 \mathrm{Mg}\) vertically from rest to \(7200 \mathrm{km} / \mathrm{h}\) in 2.0 min. You're asked to determine the required engine thrust (force) and the force exerted on a \(75-\mathrm{kg}\) astronaut during liftoff. What do you report?

Understanding the mechanism of engine thrust is crucial when designing rockets like NASA's Ares I. Thrust is the force that propels the rocket upwards, determined by Newton's third law, which states that for every action, there is an equal and opposite reaction. In simple terms, the rocket pushes exhaust gases downwards, and in turn, the gases push the rocket upwards.

For an engineer, calculating the required engine thrust involves knowing the total mass of the rocket and the desired acceleration. The thrust equation, which is a modified form of Newton's second law, incorporates both the rocket's acceleration and the acceleration due to gravity: \( T = ma + mg \). The term \( ma \) represents the force needed to accelerate the rocket while \( mg \) accounts for the force needed to overcome gravity's pull. In practice, you'd need to calculate \( a \) based on mission requirements, adding to the rocket's weight to find total thrust.

Essential to physics calculations is the ability to convert units. This skill ensures that equations are dimensionally consistent and that the computed results are meaningful. The textbook problem begins with unit conversions: the rocket's final speed from km/h to m/s and the lift-off time from minutes to seconds.

In physics, the International System of Units (SI) is the standard, with meters per second for speed, and seconds for time. Converting 7200 km/h to m/s involves multiplying by \(\frac{1}{3.6}\), a conversion factor based on the number of meters in a kilometer (1000) and the number of seconds in an hour (3600). Time conversion is more straightforward: since there are 60 seconds in a minute, multiplying by 60 turns minutes into seconds. Mastering these conversions is key to accurately applying physical laws and formulas.

Newton's second law is fundamental to understanding motion. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)). In our exercise, we apply this law to find the thrust necessary for the Ares I rocket's liftoff.

First, we computed the acceleration based on the change in velocity and time, using the formula \( a = \frac{\Delta v}{\Delta t} \). This acceleration is only due to the engines; however, when discussing thrust, we must also include gravity, which leads to the modified thrust equation mentioned previously. For the astronaut, the force they experience is directly from Newton's second law, combining the rocket's acceleration with Earth's gravity to find the total force exerted on them during liftoff.

Acceleration due to gravity, denoted as \( g \), is a constant that measures the rate of acceleration imparted to objects due to the Earth's gravitational pull. It's approximately \( 9.81 \, m/s^2 \) near the Earth's surface. This value is crucial in calculating the force needed to lift the rocket off the ground.

Gravity not only affects objects at rest but also plays a significant role in the operation of a rocket. The force to overcome gravity is a part of the total force the engines must produce. Therefore, when computing the necessary thrust for the rocket, the weight of the rocket (the product of its mass and \( g \)) adds to the force needed to achieve the desired acceleration, encompassing the intricate balance between propulsive force and gravitational force in rocket physics.