A rocket of mass 4.5 105 kg is in flight. Its thrust is directed at an angle of 45.5° above the horizontal and has a magnitude of 7.60 106 N. Find the magnitude and direction of the rockets acceleration. Give the direction as an angle above the horizonal.
To find the magnitude and direction of the rocket's acceleration, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
1. Find the horizontal and vertical components of the thrust:
The horizontal component of the thrust can be calculated as T_horizontal = T * cos(theta), where T is the magnitude of the thrust and theta is the angle of the thrust above the horizontal.
T_horizontal = 7.60 * 10^6 N * cos(45.5°) = 5.34 * 10^6 N
The vertical component of the thrust can be calculated as T_vertical = T * sin(theta), where T is the magnitude of the thrust and theta is the angle of the thrust above the horizontal.
T_vertical = 7.60 * 10^6 N * sin(45.5°) = 5.34 * 10^6 N
2. Calculate the net force in the horizontal direction:
Since there are no other horizontal forces acting on the rocket, the net force in the horizontal direction is equal to the horizontal component of the thrust.
F_net_horizontal = T_horizontal = 5.34 * 10^6 N
3. Calculate the net force in the vertical direction:
In the vertical direction, the net force is equal to the vertical component of the thrust minus the force of gravity.
F_net_vertical = T_vertical - mg, where m is the mass of the rocket and g is the acceleration due to gravity.
F_net_vertical = 5.34 * 10^6 N - (4.5 * 10^5 kg * 9.8 m/s^2) = 1.73 * 10^5 N
4. Calculate the net force:
The net force can be calculated using the Pythagorean theorem: F_net = sqrt(F_net_horizontal^2 + F_net_vertical^2)
F_net = sqrt((5.34 * 10^6 N)^2 + (1.73 * 10^5 N)^2) = 5.40 * 10^6 N
5. Calculate the acceleration:
The acceleration can be calculated using Newton's second law of motion: F_net = ma, where F_net is the net force and m is the mass of the rocket.
a = F_net / m = (5.40 * 10^6 N) / (4.5 * 10^5 kg) = 12 m/s^2
6. Calculate the angle of acceleration above the horizontal:
The angle can be calculated using the inverse tangent function: angle = atan(F_net_vertical / F_net_horizontal).
angle = atan(1.73 * 10^5 N / 5.34 * 10^6 N) = 1.88° above the horizontal.
Therefore, the magnitude of the rocket's acceleration is 12 m/s^2, and the direction of the acceleration is 1.88° above the horizontal.