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Simulation Of Projectile Motion
Literature Review
In this section, we will be including some articles relating to our project and relating them to our project about Projectile Motion.
Article 1: Projectile Motion - How it works
Science Clarified. (2010). Projectile Motion - How it works. [Online]. Available at http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Projectile-Motion-How-it-works.html. Retrieved August 24, 2015.
Summary:
With both air resistance and gravity acting on objects of different mass, such as a ball and a feather, the ball would fall faster than the feather as the air resists the feather much better.
However, under free fall conditions where gravity is the only force acting on it, objects of different mass will fall at the same rate and reach the bottom at the same time.
The path of a projectile with gravity as the only force acting on it (and air resistance taken to be negligible) follows the shape of a parabola.
The path of a projectile motion can be divided into two parts, the horizontal and vertical components, and gravity only affects the vertical motion but not the horizontal motion. As such, the horizontal velocity of the projectile (refers to the object) should be constant during the duration when the object is falling, but the horizontal velocity would be reduced to zero when its vertical velocity is zero.
How it relates to our project:
In our project, instead of a full parabola for the path of a projectile (as shown below),
we will only be taking part of a parabola (the part where the projectile falls back down to the Earth due to gravity) (as shown below).
As such, the formula for the curve of our projectile motion will look something like the normal quadratic formula, which is .
However, since we are only taking the part where the projectile falls back down to the Earth, the formula for the curve is , where the negative reflects the normal quadratic formula along the y-axis (As seen in Image 4 below), a represents the stretch of the graph, which affects the gradient of the curve (As seen in Image 5 below), b represents the translation of the graph towards the right or left by b units (As seen in Image 6 below), which can be substituted by the value of , and c represents the translation of graph upwards or downwards by c units (As seen in Image 7 below), which can be substituted with the flight altitude of the aircraft or which the bomb was released.
Image 1
Image 2
Image 3
Image 4
(How Negative affects the Graph)
Image 5
(How the value of a affects the Graph)
Image 6
(How b affects the Graph)
Image 7
(How c affects the Graph)
For our Physics model, which can be viewed under the Methodology: Physics / Math page, we have also split it into 2 components, which is the horizontal and the vertical component of the path which the bomb takes to hit the building. Using the actual horizontal distance that the bomb traveled before reaching the building, we would be able to compute the flight time of the bomb, and from the flight time, we would be able to compute the vertical distance that the bomb has to travel in order to travel the required horizontal distance to hit the building, and we check whether the bomb will have a confirm hit on the building by setting conditions which ensure the bomb hits the building.
Article 2: Galieleo's Theory of Projectile Motion
Naylor R. H. (1980). Galileo's Theory of Projectile Motion. The History of Science Society. Vol. 71 Pg 550-570. Available: http://www.jstor.org/stable/230500. Retrieved August 26 2015.
Summary:
The form of a projectile trajectory, or path of a projectile with gravity as the only force acting on it (and air resistance taken to be negligible), follows the shape of a parabola.
Galileo's Theory of Projectile Motion explains instances in which air resistance is small or negligible, and does not include instances where air resistance is significant, as he was not so interested in experimenting with projectiles of different kinds, than in obtaining precise information which could be related to a mathematical model of the trajectory (or the path of the projectile).
The path of a projectile motion can be divided into two parts, the horizontal and vertical motions which were independent. The vertical motion was considered to be unaffected by the horizontal motion of the projectile.
How it relates to our project:
The relationship between Article 2 and our project is similar to the the relationship between Article 1 and our project.
Our Physics model is split into 2 different components, which are the horizontal and vertical components of which the bomb would travel. With the actual horizontal distance the bomb travelled in order to hit the building, we would be able to compute the actual time the bomb took to hit the building from its point of release. From the computed time, we can also compute the vertical distance in which the bomb has to travel in order to travel the required horizontal distance to hit the building, and using this vertical distance, we can check if the bomb has a confirm hit on the building.
In addition, the path which our bomb takes to hit the building is a part-parabola, where the bomb would fall towards Earth at an increasing speed and constant acceleration of . The equation of the parabola in which the bomb travels to hit the building is , but the actual path the bomb takes is the part of the parabola where the value of x is more than or equal zero. a represents the stretch of the graph, which affects the gradient of the curve (As seen in Image 5 above), b represents the translation of the graph towards the right or left by b units (As seen in Image 6 above), which can be substituted by the value of , and c represents the translation of graph upwards or downwards by c units (As seen in Image 7 above), which can be substituted with the flight altitude of the aircraft or which the bomb was released.
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