A rocket of mass 4.5 105 kg is in flight. Its thrust is directed at an angle of 47.5° above the horizontal and has a magnitude of 6.60 106 N. Find the magnitude and direction of the rockets acceleration. Give the direction as an angle above the horizontal.

Its acceleration will be in the direction of the net force

Force horizontal component= Fh = 6.6*10^6 cos 47.5

Force vertical component = Fv =
6.6*10^6 sin 47.5 - 4.5*10^5*9.81

so
Fh = 4.46*10^6 N
Fv = 2.19*10^6 N

F = sqrt(Fh^2+Fv^2) = 4.97*10^6 N

tan A = Fv/Fh = 2.19/4.46

now do A = F/m

To find the magnitude and direction of the rocket's acceleration, we need to resolve the thrust force into its horizontal and vertical components.

Given:
Mass of the rocket (m) = 4.5 * 10^5 kg
Thrust magnitude (F) = 6.60 * 10^6 N
Thrust angle (θ) = 47.5° above the horizontal

Step 1: Resolve the thrust force into horizontal and vertical components.
The horizontal component of the thrust force (F_x) can be calculated using trigonometry:
F_x = F * cos(θ)

Substituting the given values:
F_x = 6.60 * 10^6 N * cos(47.5°)

Calculating F_x:
F_x ≈ 4.47 * 10^6 N

The vertical component of the thrust force (F_y) can be calculated using trigonometry:
F_y = F * sin(θ)

Substituting the given values:
F_y = 6.60 * 10^6 N * sin(47.5°)

Calculating F_y:
F_y ≈ 4.99 * 10^6 N

Step 2: Calculate the net force in the horizontal direction (F_netx).
Since there are no other horizontal forces acting on the rocket, the net horizontal force is equal to the horizontal component of the thrust force:
F_netx = F_x

Substituting the calculated value of F_x:
F_netx ≈ 4.47 * 10^6 N

Step 3: Calculate the net force in the vertical direction (F_nety).
The net vertical force (F_nety) can be calculated using Newton's second law:
F_nety = m * a

Where:
m = mass of the rocket = 4.5 * 10^5 kg
a = acceleration of the rocket

Substituting the given mass value:
F_nety = (4.5 * 10^5 kg) * a

Since the only vertical force acting on the rocket is the vertical component of the thrust force, we can equate F_nety to F_y:
F_nety = F_y

Substituting the calculated value of F_y:
(4.5 * 10^5 kg) * a = 4.99 * 10^6 N

Solving for a:
a ≈ 11.09 m/s^2

Step 4: Calculate the magnitude of the acceleration (|a|).
The magnitude of the acceleration can be calculated using the Pythagorean theorem:
|a| = sqrt(F_netx^2 + F_nety^2)

Substituting the calculated values of F_netx and F_nety:
|a| = sqrt((4.47 * 10^6 N)^2 + (4.99 * 10^6 N)^2)

Calculating |a|:
|a| ≈ 6.86 * 10^6 N

Step 5: Calculate the angle of the acceleration (α) above the horizontal.
The angle of the acceleration can be calculated using trigonometry:
α = arctan(F_nety / F_netx)

Substituting the calculated values of F_nety and F_netx:
α = arctan((4.99 * 10^6 N) / (4.47 * 10^6 N))

Calculating α:
α ≈ 47.3°

Therefore, the magnitude of the rocket's acceleration is approximately 6.86 * 10^6 N, and its direction is approximately 47.3° above the horizontal.

To find the magnitude and direction of the rocket's acceleration, we can break down the thrust force into its horizontal and vertical components.

Given:
Mass of the rocket, m = 4.5 * 10^5 kg
Thrust force, F = 6.60 * 10^6 N
Angle above the horizontal, θ = 47.5°

First, we can find the horizontal component of the thrust force (F_h):
F_h = F * cos(θ)

Substituting the given values:
F_h = (6.60 * 10^6 N) * cos(47.5°)

Next, we can find the vertical component of the thrust force (F_v):
F_v = F * sin(θ)

Substituting the given values:
F_v = (6.60 * 10^6 N) * sin(47.5°)

Now, we can calculate the acceleration using Newton's second law, F = ma.

For the horizontal direction:
F_h = m * a_h
a_h = F_h / m

Substituting the calculated value of F_h and the given value of m:
a_h = [(6.60 * 10^6 N) * cos(47.5°)] / (4.5 * 10^5 kg)

For the vertical direction:
F_v = m * a_v
a_v = F_v / m

Substituting the calculated value of F_v and the given value of m:
a_v = [(6.60 * 10^6 N) * sin(47.5°)] / (4.5 * 10^5 kg)

Now, we can calculate the magnitude of acceleration (a) using the components a_h and a_v:
a = √(a_h^2 + a_v^2)

Finally, find the direction of acceleration (θ_acc) above the horizontal using the inverse tangent function:
θ_acc = tan^(-1)(a_v / a_h)

Solve the above equations to find the magnitude and direction of the rocket's acceleration.