Simulation Of Projectile Motion
Physics Model
(Developing the Algorithm)
When an object is being released by an aircraft, it would fall towards the ground due to the Earth’s gravity pulling the object towards the core of the Earth, and an object can travel different paths as compared to other objects, depending on when the object is released, which can be seen in Diagram 1.
As such, there would be an extra horizontal distance or reduced horizontal distance between the actual point of release and the building as compared to the horizontal distance between the perfect point of release and the building, which we name Δx.
Depending on the actual point of release, Δx can be positive or negative.
Diagram 3
If the actual point of release is later than the perfect point of release, Δx would have a negative value.
Diagram 4
If the actual point of release is earlier than the perfect point of release, Δx would have a positive value.
We know that the speed / velocity of an object is equivalent to the distance travelled divided by the time taken.
In our Physics model, we would replace the v, the velocity with , the horizontal velocity of the aircraft.
t, the time taken, would be replaced with , which is the actual time taken for the bomb to hit the building, and d, the distance travelled, will be substituted with , which represents the horizontal distance between the actual point of release (of the bomb) and the building.
can be computed by adding Δx to x.
(x is the perfect horizontal distance between the building and the point of release in which the hit probability is 100%)
(Δx is the error distance between the actual point of release and the perfect point of release where the hit probability is 100%)
Diagram 6
So, by substituting the values, we get:
(Velocity = Distance divided by Time)
(Horizontal Velocity = Actual Horizontal Distance travelled by bomb divided by Actual Time taken for bomb to hit the building)
To compute the actual time taken for the bomb to hit the building from the actual point of release, we make T actual the subject of the formula:
However, if the pilot releases the bomb too early or too late, it will fall a different path as compared to the path of the bomb in which there is a 100% probability of it hitting the building.
As such, the actual vertical distance travelled by the bomb before it travels far enough horizontally to hit the building, which we call , may be different each time.
Diagram 3
In Kinematics, we learn that the distance travelled by an object under free-fall conditions where gravity is the only force acting on it can be expressed as
, where:
s
g
t
Distance travelled
Vertical acceleration due to gravity
Time Taken
And we substitute it with...
Represents
(Actual vertical distance travelled by bomb before it travels far enough to hit the building)
(Objects with gravity as the only force acting on it experience a vertical acceleration of
)
(Actual time taken for bomb to hit the building)
Our resulting equation would then be:
where has been computed in the previously using the formula:
Diagram 1
However, due to human error, the bomb may be released either earlier or later than the perfect point of release in which the hit probability is 100%.
Diagram 2
Diagram 5
However, to ensure that the bomb hits the building, the following 2 conditions must be met:
1) The actual vertical distance travelled by the bomb before it travels far enough horizontally to hit the building has to be less than or equal to the flight altitude of the aircraft which released the bomb:
If this condition is not met, the bomb would hit the ground first before reaching the building horizontally, as seen in Diagram 8.
Diagram 8
2) The actual vertical distance travelled by the bomb before it travels far enough horizontally to hit the building has to be more than or equal to the height of the building subtracted from the flight altitude of the aircraft which released the bomb:
If this condition is not met, the bomb would be too high to be able to hit the building when it travels far enough horizontally, as seen in Diagram 9.
Diagram 9
Diagram 7
Summary of Physics Model:
To compute the actual time taken for the bomb to travel the required horizontal distance in order to hit the building, we can use the following formula:
where,
refers to the actual time taken for the bomb to travel the required horizontal distance in order to hit the building.
x refers to the horizontal distance between the building and the perfect point of release where the hit probability is 100%.
Δx is the error distance between the actual point of release and the perfect point of release where the hit probability is 100%.
refers to the horizontal velocity of the aircraft which released the bomb.
To compute the actual vertical distance that the bomb has to travel in order to travel the required horizontal distance to hit the building, we can use the following formula:
where,
refers to the actual vertical distance that the bomb has to travel in order to travel the required horizontal distance to hit the building.
9.8 refers to the vertical acceleration of the bomb with gravity as the only force acting on it.
refers to the actual time taken for the bomb to travel the required horizontal distance in order to hit the building.
To ensure that the bomb hits the building successfully, the following condition must be met, or the bomb would hit the ground first before reaching the building or would be too high to be able to hit the building: