A 2 stage rocket launches off at an initial angle of 75 degrees above the horizontal. The rocket has a mass of 250 kg, while the stage 1 has a mass of 125 kg, and the stage 2 booster has a mass of 50 kg. At ignition the stage 1 booster gives the rocket a thrust of 34,652 N for 14 seconds. Then the stage 1 booster falls off and the stage 2 booster immediately fires. The stage 2 booster gives the rocket a thrust of 15,000 N for a length of 8 seconds. Then the stage 2 booster falls off and the rocket is now a projectile for the rest of its flight until it crashes into the ground. Assume the rocket went in a straight lin during the boosters firing.
a) How high did the rocket go?
b) How dar out did the rocket go?
c) How long was it in the air?
To solve this problem, we need to break it down into several steps. First, we can find the initial velocity of the rocket after the first stage.
Step 1: Calculate the acceleration during the first stage.
The thrust force provided by the stage 1 booster is 34,652 N, and the combined mass of the rocket and stage 1 is 375 kg (250 kg + 125 kg). Using Newton's second law of motion (F = ma), we can calculate the acceleration.
Acceleration during the first stage = Thrust force / Combined mass
Acceleration during the first stage = 34,652 N / 375 kg
Step 2: Calculate the initial velocity after the first stage.
The first stage booster fires for 14 seconds. Using the equation of motion (v = u + at), we can find the final velocity (u = initial velocity) after the first stage.
u = 0 (assumed at the start)
a = acceleration during the first stage (from Step 1)
t = time (14 seconds)
Initial velocity after the first stage = 0 + (Acceleration during the first stage × Time)
Step 3: Calculate the time taken by the second stage booster.
The second stage booster provides a thrust force of 15,000 N for 8 seconds. This is the time taken by the second stage booster.
Now, let's move on to answering the questions.
a) How high did the rocket go?
To find the height reached by the rocket, we need to know the time taken by the entire flight. This includes the time taken by the first stage, the time taken by the second stage, and the time the rocket spent as a projectile after both stages. We also need to find the maximum height reached during the flight.
Step 4: Calculate the time taken by the entire flight.
The entire flight time is the sum of the time taken by the first stage, the time taken by the second stage, and the time spent as a projectile.
Total flight time = Time taken by the first stage + Time taken by the second stage + Time as a projectile
Step 5: Calculate the maximum height reached.
The maximum height reached by the rocket can be calculated using the equation of motion (v^2 = u^2 + 2as).
Using s = maximum height (which we want to find),
u = initial velocity after the first stage,
v = final velocity when the rocket reaches its maximum height (which is 0),
and a = acceleration due to gravity (-9.8 m/s^2).
v^2 = u^2 + 2as
0 = (Initial velocity after the first stage)^2 + 2(-9.8) × maximum height
From this equation, we can find the maximum height reached by the rocket.
b) How far out did the rocket go?
The horizontal distance covered by the rocket can be calculated using the equation of motion (s = ut + 0.5at^2).
Using s = distance (which we want to find),
u = initial velocity after the first stage,
t = time taken by the entire flight, and
a = acceleration during the first stage (from Step 1).
From this equation, we can find the horizontal distance traveled by the rocket.
c) How long was it in the air?
We already calculated the time taken by the entire flight in Step 4. This will give us the total time the rocket was airborne.
By following these steps, you can find the answers to all three questions.