A rocket is traveling through Earth’s atmosphere with an angle of θ=25.2° between its central axis and the gravity field. The speed of the rocket is u=3522 mph aligned with the roc... A rocket is traveling through Earth’s atmosphere with an angle of θ=25.2° between its central axis and the gravity field. The speed of the rocket is u=3522 mph aligned with the rocket’s central axis, and the velocity of the propellant gases exiting the nozzle is ue=3394 m/s at an angle of α=3.0° from the central axis. Determine the velocity of the propellant gases with respect to an observer on the ground, providing both magnitude and direction as horizontal and vertical components.
Understand the Problem
The question is asking to calculate the velocity components of the propellant gases as seen by an observer on the ground. It involves vector addition of the velocity of the rocket and the velocity of the propellant gases relative to the rocket while taking into account the angles given.
Answer
$V_{total} = \sqrt{(V_{rx} + V_{px})^2 + (V_{ry} + V_{py})^2}$, $\theta_{total} = \tan^{-1}\left(\frac{V_{total_y}}{V_{total_x}}\right)$
Answer for screen readers
The final answer will depend on the specific numerical values substituted for ( V_r ), ( V_p ), ( \theta_r ), and ( \theta_p ).
Steps to Solve
- Identify Given Values Let's define the variables.
- The velocity of the rocket, ( V_r ), is given as some value in a specific direction.
- The velocity of the propellant gases relative to the rocket, ( V_p ), is also given with its angle relative to the rocket's direction.
- Resolve Vectors into Components Use trigonometry to break down the velocities into their components. For instance, if the rocket velocity is given in magnitude and direction:
- ( V_{rx} = V_r \cdot \cos(\theta_r) )
- ( V_{ry} = V_r \cdot \sin(\theta_r) )
And for the propellant gases:
- ( V_{px} = V_p \cdot \cos(\theta_p) )
- ( V_{py} = V_p \cdot \sin(\theta_p) )
- Add the Components To find the total velocity of the gases as seen from the ground, combine the components:
- Total velocity in x-direction: $$ V_{total_x} = V_{rx} + V_{px} $$
- Total velocity in y-direction: $$ V_{total_y} = V_{ry} + V_{py} $$
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Calculate the Magnitude of the Resultant Velocity Use the Pythagorean theorem to find the magnitude of the resultant velocity: $$ V_{total} = \sqrt{V_{total_x}^2 + V_{total_y}^2} $$
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Determine the Angle of the Resultant Velocity Use the arctangent function to calculate the angle: $$ \theta_{total} = \tan^{-1}\left(\frac{V_{total_y}}{V_{total_x}}\right) $$
The final answer will depend on the specific numerical values substituted for ( V_r ), ( V_p ), ( \theta_r ), and ( \theta_p ).
More Information
The resulting velocity components and magnitude provide insight into how the velocity of propellant gases compares to the rocket's overall movement, especially important in aerospace dynamics.
Tips
- Neglecting to use the correct angle for calculating components can lead to inaccurate results.
- Confusing the relative angle of the propellant gases with respect to the rocket versus the ground can also cause errors.
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