Question

A rocket with a weight of 12000 kg produces a thrust of 160kN. The rocket is launched in a vertical direction from rest, assuming no significant change in mass during this time determine its velocity and vertical height after 60 s under the following conditions: a) The air resistance is negligible. B) The air resistance produces a drag force FD=0. 3v2, where FD is in newtons and v is in metres per second

Answer 1

A)

Step-by-step explanation:

a) Assuming negligible air resistance, we can use Newton's second law of motion to determine the velocity and vertical height of the rocket after 60 seconds.

1. Velocity:

The thrust produced by the rocket is equal to the product of its mass (weight) and acceleration.

Thrust = mass * acceleration

160 kN = 12000 kg * acceleration

Solving for acceleration:

acceleration = (160 kN) / (12000 kg) = 13.33 m/s^2

Using the equation for velocity:

velocity = initial velocity + acceleration * time

Since the rocket starts from rest (initial velocity = 0):

velocity = 0 + (13.33 m/s^2) * 60 s = 800 m/s

2. Vertical Height:

We can use the equation for displacement (vertical height) in vertical motion under constant acceleration:

displacement = initial velocity * time + (1/2) * acceleration * time^2

Since the initial velocity is 0:

displacement = (1/2) * acceleration * time^2

displacement = (1/2) * (13.33 m/s^2) * (60 s)^2 = 23940 m

Therefore, after 60 seconds, assuming negligible air resistance, the rocket will have a velocity of 800 m/s and a vertical height of 23940 meters.

b) Considering air resistance with a drag force FD = 0.3v^2, we need to account for the effect of air resistance on the rocket's motion.

1. Velocity:

Using the concept of net force, we have:

Thrust - Drag force = mass * acceleration

160 kN - 0.3v^2 = 12000 kg * acceleration

At equilibrium, when the thrust is equal to the drag force, we have:

160 kN - 0.3v^2 = 0

Solving for velocity:

0.3v^2 = 160 kN

v^2 = (160 kN) / 0.3

v^2 = (160000 N) / 0.3

v^2 = 533333.33 m^2/s^2

v ≈ 730.3 m/s (approx.)

2. Vertical Height:

To determine the vertical height, we can integrate the velocity function over time:

displacement = ∫(velocity) dt

Integrating the velocity function:

displacement = ∫(730.3) dt

displacement = 730.3t + C

Since the rocket starts from rest at t = 0, the constant C is 0:

displacement = 730.3t

Substituting t = 60 s:

displacement = 730.3 * 60 = 43818 m

Therefore, after 60 seconds, considering air resistance with a drag force of FD = 0.3v^2, the rocket will have a velocity of approximately 730.3 m/s and a vertical height of 43818 meters.