By Joseph Zabinski

As an engineer, I am interested in useful applications for the laws of mathematics and physics. One application of Newton's Law of Universal Gravitation is the slingshot effect, also called gravity assist. The slingshot effect is a way in which a spacecraft can increase its speed relative to the solar system using the gravitational pull of a moon or another large body. By creating a visual model for the slingshot effect in Visual Python, I will attempt to explain how the slingshot effect works and why it is useful.

My project is a simulation of how a spacecraft would interact with the earth and the moon. Most notably, my project demonstrates the slingshot effect; however, it also shows the reverse slingshot effect, the APOLLO effect, and a spacecraft orbitting the moon. The APOLLO effect refers to a scenario in which a spacecraft begins near the earth, around the moon, and comes back towards the earth in a figure eight pattern.

My project includes two buttons which allow the user to pause and resume the simulation. My project also has a radio box which allows the user to switch between three distinct zoom types. A second radio box allows the user to select between the slingshot effect, the reverse slingshot effect, a basic orbit, and the APOLLO effect. My project also includes a graphing system that graphs the speed of the satellite with respect to the moon and the speed of the satellite with respect to the earth.

The slingshot effect is a mechanism through which a spacecraft can increase or decrease its speed relative to the solar system. The primary reason the slingshot effect is used is to reduce the large amounts of fuel that would be needed to change a spacecraft's speed and direction.

The slingshot effect works by taking a tiny fraction of a moon's kinetic energy and converting it into the spacecraft's kinetic energy. In this way, the slingshot effect obeys all the laws of mathematics and physics, including the law of conservation of energy.

**Gravity:** When a spacecraft gets close to a large body, such as a planet or a moon, the body exerts a force on the spacecraft. This force can cause the spacecraft to speed up, slow down, or change direction. This force is called gravity. The equation for the force of gravity is:

F=(GM_{1}M_{2})/r^{2}

Where F is the force acting on the on the spacecraft; G is the universal gravitational constant; M_{1} and M_{2} are the masses of the spacecraft and the moon respectively; and r is the distance from the center of mass of the spacecraft to the center of mass of the moon.

**Newton's Third Law:** Newton's third law states that for every action, there is an equal an opposite reaction. This means that the force the moon exerts on the spacecraft is equal in magnitude and opposite in direction to the force the spacecraft exerts on the moon. But if the spacecraft exerts a force on the moon, why don't we see the moon change its trajectory? The answer lies in the difference in mass between the spacecraft and the moon. The moon has so much mass compared to the spacecraft, that the spacecraft hardly affects the moon's trajectory. As a result, for the purposes of this model, I can ignore the force the spacecraft exerts on the moon.

**Reference Frames:** A reference frame is the point of view from which you observe an object or system of objects. From different reference frames, objects can appear to behave in very different ways, but the fundamental mathematics remains the same. As a result, the concept of reference frames does not affect the math or the coding of my project, but it can help explain how the slingshot effect works.

From the reference frame of the moon, the slingshot effect is very simple. The spacecraft moves toward the moon with a velocity of v_{1}. Gravity from the moon causes the spacecraft to change both speed and direction. Since gravity is a conservative force, the speed of the spacecraft as it leaves the moon's effective gravitational field will still be |v_{1}|. The only difference is the direction the spacecraft is traveling.

This image is a time-lapse diagram of the slingshot effect from the perspective of the moon. Each dot represents the spacecraft's position at an evenly-spaced time interval. As you can see, the spacecraft gains speed as it approaches the moon, but it loses speed as it moves away from the moon. As a result, the spacecrafts initial speed relative to the moon is the same as its final speed relative to the moon.

From the perspective of the solar system, the spacecraft moves toward the moon with a velocity of v_{1}. If we say that the moon's velocity relative to the solar system is v_{2}, then the spacecraft's final speed relative to the solar system will be slightly less than |v_{1}|+2|v_{2}|. This can be seen in the image above. Thus, the spacecraft can increase its speed relative to the solar system by a amount of roughly twice the speed of the moon.
The slingshot effect also works in reverse. So if the moon and the spacecraft are initially traveling in the same direction, the spacecraft can use the slingshot effect to reduce its velocity relative to the solar system by as much as twice the speed of the moon.

The realtime version of my project

The more complex version of my project