The BRIEFING
Through the use of the Coalition of Ordered Government's satellites, we have been able to identify and discover new Locust tunneling routes. Our enemy has set up a forward base of operations underground not too far from the defensive lines of Anvil Gate. As a whole, the human race has been threatened once again. Pushed back to our last efforts, Gamma Squad has been assigned the role of finding a way to destroy the target behind our lines of defense. This will both ensure the safety of our soldiers and weaken their attack. In an attempt to render our foe useless, Gamma Squad has come up with a proposal. In this particular plan, Gamma Squad wishes to launch a Class 57 Type Bomb, capable of destroying an entire city, down a ramp. In order for the bomb to acquire more speed as it falls down the ramp, Gamma Squad has attached wheels onto the bottom of it. After leaving the ramp and once launched into the air, it will follow a parabolic trajectory. Furthermore, Gamma Squad will have to figure out where it will land. In an attempt to strike our underground foe, a source of higher intelligence has given us the appropriate distance up the ramp, and the angle at which our bomb will be launched. This will provide us with a perfect velocity in which our bomb will be able to slice through the soil with ease and strike our enemy without any remorse. Moreover, the satellite in this particular region has been damaged recently and therefore is deemed useless. Due to this reason, Gamma Squad will have to discover the horizontal distance at which our bomb will land. This particular region has been said to have very soft soil, while the area around it is covered with mountains. If the bomb does not hit this particular region accurately, it could explode above ground, and thus instigate a savage attack from the Locust onto Anvil Gate. To replicate this very action, Gamma Squad has been assigned the task to create a ramp 85cm upward and 40 degrees above the horizontal. Using this very structure, Gamma Squad will utilize a toy car to replace the actual bomb. This modeled demonstration will help the COG in finding the velocity in which to launch the bomb, and the landing zone of the bomb. Remember soldiers, a dead Locust is a safe Locust! For all civilians and fellow soldiers interested in this news and intend to be kept up to date, please follow us on our blog.
Apparatus Setup
Below are a few pictures showing our apparatus information/setup:
Variables
Given Variables:
Gamma Squad's Higher Intelligence Information:
Distance Up Ramp = 85.0 cm
Angle of Ramp = 40.0 degrees (from the horizontal)
Vertical falling distance of car = 0.795 m (The Height of the Counter Top - The Height of the Textbooks)
The Vertical Component:
A = -9.8 m/s^2
D = 0.795 m (The Height of the Counter Top - The Height of the Textbooks)
Vi = ? (Need in Order to Evaluate for T)
T = ?
The Horizontal Component:
A = 0 m/s^
D = ? (Need for the Landing Spot of the Car)
T = ? (Need From the Vertical Component in Order to Solve for the Distance)
The Dependent Variables:
The Vertical Component:
Vi = Dependent Upon the Angle and the Distance Up the Ramp
T = Dependent Upon the Variables: A, D, and Vi
The Horizontal Component:
T = Dependent Upon the Time in the Vertical Component
D = Dependent Upon the Variables: T, Vi, and A; this is the horizontal displacement that the car undergoes
The Independent Variables:
The Vertical Component:
A = -9.8 m/s^2 (Independent, and No Other Variables Affect it as Gravity is Produced by a Larger Body Mass than Any Object Artificially Made)
The Horizontal Component:
A = 0 (Independent, and No Other Variables Affect it as Gravity is Produced by a Larger Body Mass than Any Object Artificially Made)
The Controlled Variables:
The Vertical Component:
The height of the ramp had to be held constant in order to receive the initial velocity, and accurate motion of the car.
The height of the textbooks (2.75 cm) was constant in order to evaluate the vertical distance and accurately predict the landing zone of the car.
The Horizontal Component:
The angle of the ramp was to be held constant in an attempt to receive the precise velocity and accurate landing zone of the car.
The Constant Variables:
The Vertical Component:
The acceleration of the vertical component was always constant as gravity never changes.
The height if the table never changed as the tables used for this particular lab did not change physically.
The Horizontal Component:
The acceleration of the horizontal component was always constant as gravity does not affect the horizontal velocity of a moving object.
The horizontal velocity in our particular lab never changed as gravity does not decrease or increase the value of horizontal velocity.
Gamma Squad's Higher Intelligence Information:
Distance Up Ramp = 85.0 cm
Angle of Ramp = 40.0 degrees (from the horizontal)
Vertical falling distance of car = 0.795 m (The Height of the Counter Top - The Height of the Textbooks)
The Vertical Component:
A = -9.8 m/s^2
D = 0.795 m (The Height of the Counter Top - The Height of the Textbooks)
Vi = ? (Need in Order to Evaluate for T)
T = ?
The Horizontal Component:
A = 0 m/s^
D = ? (Need for the Landing Spot of the Car)
T = ? (Need From the Vertical Component in Order to Solve for the Distance)
The Dependent Variables:
The Vertical Component:
Vi = Dependent Upon the Angle and the Distance Up the Ramp
T = Dependent Upon the Variables: A, D, and Vi
The Horizontal Component:
T = Dependent Upon the Time in the Vertical Component
D = Dependent Upon the Variables: T, Vi, and A; this is the horizontal displacement that the car undergoes
The Independent Variables:
The Vertical Component:
A = -9.8 m/s^2 (Independent, and No Other Variables Affect it as Gravity is Produced by a Larger Body Mass than Any Object Artificially Made)
The Horizontal Component:
A = 0 (Independent, and No Other Variables Affect it as Gravity is Produced by a Larger Body Mass than Any Object Artificially Made)
The Controlled Variables:
The Vertical Component:
The height of the ramp had to be held constant in order to receive the initial velocity, and accurate motion of the car.
The height of the textbooks (2.75 cm) was constant in order to evaluate the vertical distance and accurately predict the landing zone of the car.
The Horizontal Component:
The angle of the ramp was to be held constant in an attempt to receive the precise velocity and accurate landing zone of the car.
The Constant Variables:
The Vertical Component:
The acceleration of the vertical component was always constant as gravity never changes.
The height if the table never changed as the tables used for this particular lab did not change physically.
The Horizontal Component:
The acceleration of the horizontal component was always constant as gravity does not affect the horizontal velocity of a moving object.
The horizontal velocity in our particular lab never changed as gravity does not decrease or increase the value of horizontal velocity.
Equations And Data Collected
Below are the equations we used in order to determine the horizontal displacement of the can.
It must be noted that this is a two part problem. The first part involved finding the final velocity of the car on the ramp, which is the initial velocity used in the second part (which involved finding the horizontal displacement of the actual can).
For the first part, we use an equation that we are fairly familiar with, showing the final velocity that can be calculated for free fall:
It must be noted that this is a two part problem. The first part involved finding the final velocity of the car on the ramp, which is the initial velocity used in the second part (which involved finding the horizontal displacement of the actual can).
For the first part, we use an equation that we are fairly familiar with, showing the final velocity that can be calculated for free fall:
However, the car will be on a ramp and not free falling. The force of gravity is still acting on the car and causing it to accelerate, but at a different, non vertical rate. We use (g)*(sin)(theta) to figure out the final velocity on the ramp, instead of just (g):
To determine the horizontal displacement of the car, we must figure out the time it takes for the car to fall the vertical displacement. Note that the horizontal stretch that the car traveled (on the counter) was not included in any of our calculations because the speed remained fairly constant over this time (or so we thought!), and we thus used to final velocity at the bottom of the ramp in our following calculations:
The final velocity from the previous part is now used to calculate horizontal displacement using the above time:
Thus we placed the can at a distance of roughly 132 cm from the counter. The can had diameter and should therefore have been large enough to accommodate for slight calculation errors, but our car unfortunately completely missed the can.
Method
For our apparatus setup, we were given many materials including, the wooden ramp, orange tracks, retort stand and clamps, metre sticks, and a stop watch.
To set up the apparatus, the wooden board was placed in between two retort stands and it was adjusted according to our measurements, which placed the track 40 degrees from the horizontal and placed the car 85 cm up the track.
The angle was properly achieved by using trigonometry, not a protractor. If the angle was 40 degrees and the distance up the track (hypotenuse) was 85 cm, then we could calculate the height of the retort stands by using the sine function.
After finding the height, we used tape to attach the track to the table/wooden board so it would not move during the experiment (however, we later realized that this slightly slowed the car down).
We made sure that the track was horizontal after the slanted ramp area, which would keep the car's velocity fairly constant (with very little friction), meaning the final velocity of the car leaving the ramp in our calculations would be about the same.
We then proceeded to obtain the final velocity of the car leaving the ramp through our calculations (seen below), and then figure out the distance the car would travel after leaving the counter.
The can (target) was placed at this distance, and our car was dropped from 85 cm up the ramp to complete failure.
To set up the apparatus, the wooden board was placed in between two retort stands and it was adjusted according to our measurements, which placed the track 40 degrees from the horizontal and placed the car 85 cm up the track.
The angle was properly achieved by using trigonometry, not a protractor. If the angle was 40 degrees and the distance up the track (hypotenuse) was 85 cm, then we could calculate the height of the retort stands by using the sine function.
After finding the height, we used tape to attach the track to the table/wooden board so it would not move during the experiment (however, we later realized that this slightly slowed the car down).
We made sure that the track was horizontal after the slanted ramp area, which would keep the car's velocity fairly constant (with very little friction), meaning the final velocity of the car leaving the ramp in our calculations would be about the same.
We then proceeded to obtain the final velocity of the car leaving the ramp through our calculations (seen below), and then figure out the distance the car would travel after leaving the counter.
The can (target) was placed at this distance, and our car was dropped from 85 cm up the ramp to complete failure.
Data Analysis
Ultimately, the can was placed too far from where the car should have fallen in.
This, we realized, was due to our vertical displacement calculation missing a key component; the height of the can. We can analyze that our car would have exactly hit the textbooks, but the can was placed on these textbooks and our vertical displacement should have subtracted the height of the can, giving the car a safe landing area after our calculations were made.
The analysis of our data is explained further under Conclusion.
This, we realized, was due to our vertical displacement calculation missing a key component; the height of the can. We can analyze that our car would have exactly hit the textbooks, but the can was placed on these textbooks and our vertical displacement should have subtracted the height of the can, giving the car a safe landing area after our calculations were made.
The analysis of our data is explained further under Conclusion.
Media of Testing
Please proceed to watching the disastrous final test of Operation Gamma : Orange Road*:
Conclusion: Interpretation, Error Analysis, Improvements
Throughout the course of the lab, a few unnoticeable mistakes lead to our hot wheels car narrowly missing our target; the can inside the box. Firstly, we didn’t take into account the height of the can and the box it was inside while plugging in values into our derived equation. We had realized this after acknowledging the fact that the car, without any interference from the box or can, would have landed on top of our textbooks and be very close to our predicted landing zone. However, we did not take into account the height of the can which the car had to fall in, and therefore, the car struck the top of the box and failed to reach its target. At the end of our lab, our evaluated equation was successful in telling us only the distance the can had to be from the ramp. If we had taken in account the height of the can and the box, we would most likely have been successful. Another mistake that could have been avoided during the lab was putting tape on the ramp. We can assume that the tape slowed down the velocity of the car and may not have given the car enough speed to reach our target. Nevertheless, this information is the least detrimental to our observations as it would have only slowed the car down slightly, with the decrease in speed being negligible. If we were to do this experiment again, we would re-evaluate the variables in our equation (particularly the vertical displacement of the car after leaving the track), and remove the tape from both sides of the ramp. By doing this, we would have higher chances of getting the car into the can.
When launching the Class 52 Bomb during the operations of Orange Road*, we therefore did not take into account the height of the surrounding mountains. The bomb came into contact with one of the mountains and clipped its right side. The bomb spun uncontrollably in the air, and totally missed our target. Instead, it destroyed much of our terrain that we had used defensively for Anvil Gate. Upon hearing the explosion, the Locust emerged out, and with a fear of another strike, destroyed our headquarters at Anvil Gate. Our defenses were overrun in a matter of seconds, as they had tanks that broke right through the gate and destroyed the city within. We have just lost one of our most precious cities as it held significant amounts of fuel and food. All of our soldiers and civilians of Anvil Gate will be missed. It must be noted that the COG is very disappointed in Gamma Squad, and hereby demotes you to a lower rank of expertise. Your services will no longer be required for situations that require such essential success. Thank you for your efforts.
When launching the Class 52 Bomb during the operations of Orange Road*, we therefore did not take into account the height of the surrounding mountains. The bomb came into contact with one of the mountains and clipped its right side. The bomb spun uncontrollably in the air, and totally missed our target. Instead, it destroyed much of our terrain that we had used defensively for Anvil Gate. Upon hearing the explosion, the Locust emerged out, and with a fear of another strike, destroyed our headquarters at Anvil Gate. Our defenses were overrun in a matter of seconds, as they had tanks that broke right through the gate and destroyed the city within. We have just lost one of our most precious cities as it held significant amounts of fuel and food. All of our soldiers and civilians of Anvil Gate will be missed. It must be noted that the COG is very disappointed in Gamma Squad, and hereby demotes you to a lower rank of expertise. Your services will no longer be required for situations that require such essential success. Thank you for your efforts.