A rocket of mass 4.45E+5 kg is in flight. Its thrust is directed at an angle of 51.1° above the horizontal and has a magnitude of 7.64E+6 N. Calculate the magnitude (enter first) and direction of the rocket's acceleration. Give the direction as an angle above the horizontal.
To calculate the magnitude and direction of the rocket's acceleration, we can use Newton's second law of motion.
First, we need to resolve the thrust force into its horizontal and vertical components. The horizontal component of the thrust can be calculated using the equation:
Thrust(horizontal) = Thrust * cos(angle)
Substituting the given values:
Thrust(horizontal) = 7.64E+6 N * cos(51.1°)
Next, we can calculate the vertical component of the thrust using the equation:
Thrust(vertical) = Thrust * sin(angle)
Substituting the given values:
Thrust(vertical) = 7.64E+6 N * sin(51.1°)
Now, since the rocket is in flight, we need to consider the forces acting on it. The only force acting on the rocket is the thrust, which can be considered as the net force.
Applying Newton's second law, the net force is equal to the mass of the rocket multiplied by its acceleration:
Net force = Mass * Acceleration
Substituting the given values and separating the forces into horizontal and vertical components:
Net force(horizontal) = Mass * Acceleration(horizontal)
Net force(vertical) = Mass * Acceleration(vertical)
Now, we can equate the net forces to the corresponding components of the thrust:
Net force(horizontal) = Thrust(horizontal)
Net force(vertical) = Thrust(vertical)
By comparing the corresponding components, we can solve for the acceleration:
Acceleration(horizontal) = Thrust(horizontal) / Mass
Acceleration(vertical) = Thrust(vertica) / Mass
Substituting the values, we get:
Acceleration(horizontal) = (7.64E+6 N * cos(51.1°)) / 4.45E+5 kg
Acceleration(vertical) = (7.64E+6 N * sin(51.1°)) / 4.45E+5 kg
Finally, we can use the magnitude and direction of the acceleration to calculate the resultant magnitude and direction.
Magnitude of acceleration = sqrt((Acceleration(horizontal))^2 + (Acceleration(vertical))^2)
Direction of acceleration = atan(Acceleration(vertical) / Acceleration(horizontal))
Now, let's calculate the magnitude and direction of the rocket's acceleration.
First, let's calculate the horizontal and vertical components of the thrust:
Thrust(horizontal) = 7.64E+6 N * cos(51.1°) ≈ 4.89E+6 N
Thrust(vertical) = 7.64E+6 N * sin(51.1°) ≈ 5.75E+6 N
Next, let's calculate the horizontal and vertical components of the acceleration:
Acceleration(horizontal) = (4.89E+6 N) / (4.45E+5 kg) ≈ 11 m/s^2
Acceleration(vertical) = (5.75E+6 N) / (4.45E+5 kg) ≈ 12.93 m/s^2
Now, let's calculate the magnitude and direction of the acceleration:
Magnitude of acceleration = sqrt((11 m/s^2)^2 + (12.93 m/s^2)^2) ≈ 17 m/s^2
Direction of acceleration = atan(12.93 m/s^2 / 11 m/s^2) ≈ 49.7° above the horizontal
Therefore, the magnitude of the rocket's acceleration is approximately 17 m/s^2, and its direction is approximately 49.7° above the horizontal.