9. The rocket engine of a 10 tonne missile produces a thrust of 400 kN. The missile is launched in a vertical direction. The air resistance is 0.75v2 N where v is the speed of the missile.
(i) Find the speed of the missile after one minute.
(ii) Find the distance travelled by the missile after one minute.
F = ma
(i) v = at
(ii) s = 1/2 at^2
(i) To find the speed of the missile after one minute, we need to consider the net force acting on the missile.
The weight of the missile is given by:
Weight = mass x acceleration due to gravity
Weight = 10 tonnes x 9.8 m/s^2 (1 tonne = 1000 kg)
Weight = 98,000 N
The thrust force of the rocket engine is 400 kN (400,000 N), acting in the upward direction.
The air resistance is given by 0.75v^2 N, where v is the speed of the missile.
The net force acting on the missile is the sum of the thrust force, weight force, and the negative air resistance force:
Net force = Thrust - Weight - Air resistance
At equilibrium, the net force is 0, so we can set up the equation:
0 = Thrust - Weight - Air resistance
0 = 400,000 N - 98,000 N - 0.75v^2 N
Simplifying the equation:
0.75v^2 = 302,000 N
v^2 = 302,000 N / 0.75
v^2 = 402,667 N
v ≈ √402,667 N
v ≈ 634.6 m/s (rounded to one decimal place)
Therefore, the speed of the missile after one minute is approximately 634.6 m/s.
(ii) To find the distance traveled by the missile after one minute, we need to calculate the displacement.
Displacement can be calculated using the formula:
Displacement = Initial velocity x Time + (1/2) x Acceleration x Time^2
In this case, the initial velocity is 0 m/s because the missile starts from rest.
The acceleration is the net force divided by the mass of the missile:
Acceleration = (Thrust - Weight - Air resistance) / mass
Acceleration ≈ (400,000 N - 98,000 N - 0.75v^2 N) / 10,000 kg
Acceleration ≈ (302,000 N) / 10,000 kg
Acceleration ≈ 30.2 m/s^2
Using the given time of one minute (60 seconds), we can calculate the displacement as follows:
Displacement = 0 x 60 + (1/2) x 30.2 m/s^2 x (60 s)^2
Displacement ≈ 0 + 0.5 x 30.2 m/s^2 x 3600 s^2
Displacement ≈ 0 + 0.5 x 30.2 m/s^2 x 3600 s^2
Displacement ≈ 0 + 543,600 m
Displacement ≈ 543,600 m
Therefore, the distance traveled by the missile after one minute is approximately 543,600 meters.
To find the speed of the missile after one minute, we need to use the concept of Newton's second law of motion and the principle of equilibrium. The net force acting on the missile can be found using the equation:
Net Force = Thrust - Air Resistance
Given that the thrust of the rocket engine is 400 kN and the air resistance is 0.75v^2 N, we can write the equation as:
Net Force = 400 kN - 0.75v^2 N
Now, we know that according to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the mass of the missile is given as 10 tonnes or 10,000 kg. The acceleration can be calculated using the following equation:
Acceleration = Net Force / Mass
Substituting the values, we get:
Acceleration = (400 kN - 0.75v^2 N) / 10,000 kg
Now, we need to solve this equation for the speed, v. However, the equation is not in a form that can be directly solved. We can convert 400 kN into N and then rearrange the equation to isolate the variable v. Here's how:
1 kN = 1000 N (since 1 kN is equal to 1000 Newtons)
400 kN = 400,000 N
Plugging this value back into the equation, we get:
Acceleration = (400,000 N - 0.75v^2 N) / 10,000 kg
Now we can rearrange the equation to solve for v. Here's the equation again:
Acceleration = (400,000 N - 0.75v^2 N) / 10,000 kg
Multiplying both sides of the equation by 10,000 kg, we get:
Acceleration * 10,000 kg = 400,000 N - 0.75v^2 N
Now rearranging the equation, we get:
0.75v^2 N = 400,000 N - Acceleration * 10,000 kg
Next, we isolate v^2 by subtracting 400,000 N from both sides:
0.75v^2 N - 400,000 N = - Acceleration * 10,000 kg
Finally, we divide by 0.75 N to isolate v^2:
v^2 = (- Acceleration * 10,000 kg + 400,000 N) / 0.75 N
Now we can find v by taking the square root of both sides:
v = √((- Acceleration * 10,000 kg + 400,000 N) / 0.75 N)
Plugging in the given values and calculating, we find the speed of the missile after one minute.