- Calculate the net force by subtracting weight from thrust: \(3.3 \times 10^7 \, \text{N} - 2.8 \times 10^7 \, \text{N} = \boxed{5,000,000 \, \text{N}}\).
- Determine mass by dividing weight by gravity: \(2.8 \times 10^7 \, \text{N} / 9.8 \, \text{m/s}^2 \approx \boxed{2,857,142.86 \, \text{kg}}\).
- Find acceleration by dividing net force by mass: \(5,000,000 \, \text{N} / 2,857,142.86 \, \text{kg} \approx \boxed{1.75 \, \text{m/s}^2}\).
Explanation
- Analysis of Rocket Dynamics
We are analyzing the dynamics of a rocket at launch. We are given the weight of the rocket \(W = 2.8 \times 10^7 \, \text{N}\) and the thrust produced by its engines \(F_{\text{thrust}} = 3.3 \times 10^7 \, \text{N}\). The acceleration due to gravity is taken as the standard value \(g = 9.8 \, \text{m/s}^2\). Our objective is to determine the net force acting on the rocket, the mass of the rocket, and the resulting acceleration at launch. To do this, we use the principles of classical mechanics: the net force is the difference between the upward thrust and downward weight, the mass is derived from weight via gravity, and acceleration is calculated using Newton's second law, \(F_{\text{net}} = m \cdot a\).
- Calculating Net Force
The net force \(F_{\text{net}}\) acting on the rocket is the difference between the upward thrust force and the downward force of gravity (the weight).
The calculation is:
\[F_{\text{net}} = F_{\text{thrust}} - W\]
\[F_{\text{net}} = 3.3 \times 10^7 \, \text{N} - 2.8 \times 10^7 \, \text{N} = 0.5 \times 10^7 \, \text{N} = 5,000,000 \, \text{N}\]
Thus, the net force acting on the rocket is \(5,000,000 \, \text{N}\).
- Calculating Mass
The mass \(m\) of the rocket is related to its weight \(W\) and the gravitational acceleration \(g\) by the formula \(W = m \cdot g\), which implies \(m = \frac{W}{g}\).
Using the given values:
\[m = \frac{2.8 \times 10^7 \, \text{N}}{9.8 \, \text{m/s}^2}\]
\[m \approx 2,857,142.86 \, \text{kg}\]
So, the mass of the rocket is approximately \(2,857,142.86 \, \text{kg}\).
- Calculating Acceleration
According to Newton's second law of motion, the net force \(F_{\text{net}}\) equals the product of the rocket's mass \(m\) and its acceleration \(a\). Rearranging this for acceleration gives \(a = \frac{F_{\text{net}}}{m}\).
Substituting our previously calculated values:
\[a = \frac{5,000,000 \, \text{N}}{2,857,142.86 \, \text{kg}}\]
\[a \approx 1.75 \, \text{m/s}^2\]
Therefore, the upward acceleration of the rocket is \(1.75 \, \text{m/s}^2\).
Examples
Understanding rocket dynamics is a fundamental part of aerospace engineering. By calculating the net force and acceleration, engineers ensure that a rocket can successfully overcome gravity to reach orbit. For instance, when designing a launch vehicle, engineers must precisely determine the fuel weight versus thrust to maintain a stable, effective ascent trajectory, much like the one calculated here. This math is the foundation for everything from satellite deployment to space exploration missions.