3, 2, 1 … Time to lift off…
On 30 May 2020, it was indeed a remarkable event of the successful launch of the SpaceX Falcon9 rocket. On the live video, it was showing a live event about the launching of SpaceX Falcon9 rocket, and displaying the speed of the rocket over time as well, which I found their figures were very interesting.
I am a small fan of SpaceX program and I have been following them once a while, so certainly I won’t miss this great achievement by SpaceX.
Today I am going to discuss about the speed and time of the SpaceX Falcon9 rocket. I had spent some time to record the speed and the time that were displayed from the live video.
This is how I did it.
For every 10 seconds, I had recorded the speed and the altitude.
I had recorded the SpaceX Falcon9 rocket speed and time starting from 20 seconds onwards, because the live video started to display the time around 16 seconds after the rocket launched.
At 20 seconds, the speed was ~272km/h and altitude was 0.7 km.
At 30 seconds, the speed was ~459km/h and altitude was 1.7 km.
At 40 seconds, the speed was ~682km/h and altitude was 3.2 km.
I had repeated the same process for the rest of the time.
I had keyed in those data into the excel sheet. Table 1 below shows the data starting from 20 seconds to 40 seconds.
Table 1.
Next, I convert the speed in (km/h) to (m/s) which is SI unit.
I convert 272 km/h to m/s by using the method below
Table 2.
After I had converted all the units, I used the excel sheet to generate the speed-time graph of the SpaceX Falcon9 rocket.
The X-axis is time in seconds, the Y-axis is speed in m/s.
Let me analyze what was happening at the point A, point B and point C.
At point A (time ~160s), there was a slight decrease in speed. It was because at time ~160s, the rocket had reached an important stage, where the first stage propellant was being separated from the rocket. In order to have a smooth and safe separation, the first stage propellant need to be turned off, and the second stage propellant can only be ignited after the first stage propellant had been completely separated from the rocket. Since there were no ignitions during this separation stage, therefore the rocket lost the required upward force. This upward force was initially provided for the rocket to escape from the earth. At the same time, the force of gravity was pulling the rocket back to the earth, the resultant force was therefore facing towards earth. This downward resultant force was the force that caused the decrease in speed of the rocket.
Refer to the Table 3 above, from 157 seconds onwards, the speed of the rocket was decreasing, thus it is shown a negative acceleration. At 165 seconds, as the stage 2 propellant started the ignition, the rocket started to gain upward force and gradually resist the force of gravity. From 170 seconds onwards, the rocket started to gain positive acceleration and was speeding up.
At point B (time ~360s), the rocket had reached the altitude destination (200km above the earth). Notice that the speed was still continuing to increase, which also means that the rocket was still accelerating. It was because the speed of the rocket was still not reached the required orbital velocity yet. Thus the rocket needs to continue accelerating.
At point C (time ~540s), notice that the speed had stop increasing. This was happened when the rocket turns off its second stage propeller engine. Notice that the speed is ~7500 m/s. When the rocket had reached the required orbital velocity (~7500 m/s) at the 200km above the earth, it had no longer required the ignition to keep it in the orbit. This is because the orbital velocity of the rocket is able to counterbalance the pull of earth gravity, which enable the rocket maintains its orbit at 200km above the earth.
Next, I had calculated the acceleration based on the speed-time graph, recalled that the acceleration definition:
t0 = initial time
t1 = sometime after t0
V0 = Speed at time t0
V1 = Speed at time t1
To calculate the acceleration in between 20 seconds and 30 seconds, when t0 = 20s, V0 = 75.56m/s; when t1 = 30s, V1 = 127.50 m/s,
Table 4.
I had repeated the same process for the rest of the time.
After I had calculated all the accelerations, I used the excel sheet to plot the acceleration versus time graph.
The X-axis is time in second and the Y-axis is acceleration in m/s2.
At point A (~60s), notice that the acceleration was decreasing. Please take note that decreasing in acceleration does not necessarily mean a decrease in velocity, as long as acceleration remain in positive value, the object is still accelerating and gaining speed. Refer to the figure below, as the acceleration was decreasing, the speed of the rocket was still increasing.
The reason why the acceleration was decreasing at point A was because the rocket had reached to the maximum dynamic pressure (Max q). The dynamic pressure is directly proportionally to air density and speed of the object. Therefore, the higher the speed of the rocket, the greater the dynamic pressure that acts on the surface of the rocket. For every rocket, there is a maximum pressure that the rocket can withstand, so when the rocket reaches the maximum dynamic pressure stage, the thrust force of the propeller engine is being reduced in order to decrease the speed of the rocket, so that the rocket will not be torn apart by the dynamic pressure.
At point B (~160s), the separation of the first stage propellant occurred. Since the propeller engine was still not activated yet, there was no upward force to provide the thrust that was needed by the rocket, and the pull of earth gravity was starting to pull the rocket. Thus, there was a sudden drop in acceleration. At this time, we can say that the rocket speed was decreasing, hence it is showing a negative acceleration.
At point C (~540s), the speed of the rocket had reached orbital velocity and the rocket turned off its propeller engine. Notice that the acceleration had maintained at 0 m/s2 after 540s, it’s shown that the speed of the rocket neither increase nor decrease. It’s also shown that the orbital velocity of the rocket was able to counterbalance the force of gravity, even though the propeller engine was turned off.
From the speed-time graph and the acceleration graph, I estimate the mass of the rocket during initial phase.
Recalled the Force formula:
Assume that the g is constant across all the different altitudes.
From the Eq2, the thrust force generated by rocket propellant has to be much greater than the weight of the rocket in order to create a tremendous upward resultant force.
We can then use some mathematics work out:
I had collected one key information from Wikipedia, which is the initial thrust force = 7,607 kN, and from Table 5, the acceleration is 3.78 m/s2, thus, the estimated mass of the rocket is calculated by using Eq 5.
*Important note: according to Wikipedia, the mass of the rocket is 549054 kg, all the values are based on estimation. The accuracy of the result is not the key focus in today’s discussion, the key focus here is to understand the concept of the resultant force. When two forces are acting on the object and both of the forces are in opposite directions to each other, the resultant force is just the result of summation of 2 vector numbers. In this case, the Fg is downward force, and Ft is the upward force, so both of them are in opposite directions. Assume that a positive vector number representing the upward direction and a negative vector number representing the downward direction, then Ft is a positive vector number, and the Fg is a negative vector number. If Ft is greater than the Fg, hence the summation of Ft and Fg produces a positive vector number of resultant force F, which indicates the rocket is flying upward direction. Otherwise, if the Ft is lower than the Fg, hence the summation of Ft and Fg produce a negative vector number of resultant force F, the rocket will stay on the ground because the thrust force is not great enough to overcome the force of gravity.
It is very difficult to calculate the mass of the rocket for subsequent stages of the rocket launch based on the speed-time graph. It is because the thrust force was changing across the entire rocket launching process, and as the fuel that generate the thrust was kept burning and consumed over time, thus the mass of the rocket would reduce accordingly.
Interesting facts about the SpaceX Falcon9 rocket:
- The SpaceX Falcon9 rocket is a 2 stage rocket. There was second stage propeller engine separation occurred at time ~730 seconds after rocket had launched.
- SpaceX had developed several versions of the rocket, they were Falcon1, Falcon5 and Falcon9. The number behind the Falcon indicates the total number of thrust engines (Merlin engine) that being used in the first stage propellant.
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