A rocket of mass 3.45x10^5 kg isn’t flight near earth surface. it’s the rest is directed at an angle of 52.2° above the horizontal and has a magnitude of 8.52x10^6 N. Find the (a) magnitude and (b) direction of the rockets acceleration. give the direction as an angle above the horizontal.
To find the magnitude and direction of the rocket's acceleration, we can begin by resolving the rocket's weight into horizontal and vertical components.
Step 1: Find weight (W)
The weight of the rocket is given by the formula:
W = mg
where m is the mass of the rocket (3.45x10^5 kg) and g is the acceleration due to gravity (typically 9.8 m/s^2 near the Earth's surface). Let's calculate the weight:
W = (3.45x10^5 kg) * (9.8 m/s^2)
W ≈ 3.381x10^6 N
Step 2: Resolve the weight
We resolve the weight into two components: one in the vertical direction and another in the horizontal direction.
The vertical component of the weight (Wv) is given by the formula:
Wv = W * sinθ
where θ is the angle above the horizontal (52.2°). Let's calculate Wv:
Wv = (3.381x10^6 N) * sin(52.2°)
Wv ≈ 2.496x10^6 N
The horizontal component of the weight (Wh) is given by the formula:
Wh = W * cosθ
where θ is the angle above the horizontal (52.2°). Let's calculate Wh:
Wh = (3.381x10^6 N) * cos(52.2°)
Wh ≈ 2.184x10^6 N
Step 3: Calculate the net force
The net force acting on the rocket is the sum of the given force and the horizontal component of the weight.
The net force (F_net) is given by the formula:
F_net = F_applied + Wh
where F_applied is the given force (8.52x10^6 N) and Wh is the horizontal component of the weight. Let's calculate F_net:
F_net = (8.52x10^6 N) + (2.184x10^6 N)
F_net ≈ 1.1376x10^7 N
Step 4: Calculate acceleration
Using Newton's second law of motion, we can calculate the acceleration (a) of the rocket.
The equation is:
F_net = ma
where m is the mass of the rocket. Rearranging the equation, we can solve for a:
a = F_net / m
Let's calculate a:
a = (1.1376x10^7 N) / (3.45x10^5 kg)
a ≈ 32.992 m/s^2
(a) The magnitude of the rocket's acceleration is approximately 32.992 m/s^2.
Step 5: Calculate direction
To find the direction of the acceleration, we need to determine the angle above the horizontal.
The angle above the horizontal (θ_acc) is given by the formula:
θ_acc = arctan(Wv / Wh)
where Wv is the vertical component of the weight and Wh is the horizontal component of the weight. Let's calculate θ_acc:
θ_acc = arctan((2.496x10^6 N) / (2.184x10^6 N))
θ_acc ≈ 50.04°
(b) The direction of the rocket's acceleration is approximately 50.04° above the horizontal.
Therefore, the answers are:
(a) The magnitude of the rocket's acceleration is approximately 32.992 m/s^2.
(b) The direction of the rocket's acceleration is approximately 50.04° above the horizontal.