91Ó°ÊÓ

To comprehend how a rocket propels through space, we must understand thrust force. In essence, thrust is the force that moves a rocket forward, originating from the expulsion of fuel. For our particular problem, the thrust force \( F_T \) can be determined using the formula:\[ F_T = \dot{m} \cdot v_e \]- \( \dot{m} \) stands for the rate at which fuel is consumed, given as 120 kg/s.- \( v_e \) represents the exhaust velocity, which is 640 m/s.By multiplying these two values, we obtain the thrust force:\[ F_T = 120 \, \text{kg/s} \times 640 \, \text{m/s} = 76800 \, \text{N} \]This tells us that the rocket is experiencing a thrust force of 76800 Newtons, pushing it forward. This force is essential for overcoming the gravitational pull and guiding the rocket along its intended trajectory.

When rockets ascend, they fight against gravity, which pulls everything towards Earth. Given the trajectory angle of the rocket is 30° from the vertical, it is crucial to decompose gravity into components that align with this trajectory. The vertical component \( F_g \) of gravitational force is calculated as:\[ F_g = m \cdot g = 2800 \, \text{kg} \times 9.34 \, \text{m/s}^2 = 26152 \, \text{N} \]To comprehend how gravity affects the rocket, decompose \( F_g \) into normal \( F_{g_n} \) and tangential \( F_{g_t} \) components:- Normal component is calculated as: \[ F_{g_n} = F_g \cdot \cos(\theta) = 26152 \, \text{N} \times \cos(30^\circ) = 22630 \, \text{N} \]- Tangential component is: \[ F_{g_t} = F_g \cdot \sin(\theta) = 26152 \, \text{N} \times \sin(30^\circ) = 13076 \, \text{N} \]These components guide us in understanding how gravity works against the rocket, with \( F_{g_n} \) opposing lift and \( F_{g_t} \) affecting the direction of travel.

Newton's second law provides a framework for understanding movement under the action of forces. It tells us that the acceleration \( a \) of an object is directly proportional to the net force \( F \) acting on it and inversely proportional to its mass \( m \):\[ F = ma \]In our scenario, utilizing Newton's second law helps determine the rocket's acceleration along two primary directions:- **Normal component** of acceleration: - The net force in this direction: \[ F_{net_n} = -F_g \cdot \cos(\theta) = -22630 \, \text{N} \] - Acceleration derived using: \[ a_n = \frac{F_{net_n}}{m} = -\frac{22630 \, \text{N}}{2800 \, \text{kg}} = -8.08 \, \text{m/s}^2 \]- **Tangential component** of acceleration: - Net force calculation: \[ F_{net_t} = F_T - F_g \cdot \sin(\theta) = 76800 \, \text{N} - 13076 \, \text{N} \] - Resultant acceleration: \[ a_t = \frac{F_{net_t}}{m} = \frac{63724 \, \text{N}}{2800 \, \text{kg}} = 22.69 \, \text{m/s}^2 \]Through these calculations, Newton's law allows us to predict the precise acceleration of the rocket, crucial for fine-tuning its path and ensuring its safe ascent.