SLINGSHOT PROJECT By Dale Henderson Jarly López Robert Kemp Stacey Cotty

New Horizons Our project is based on the New Horizons mission. This mission, (to be launched in 2006) is going to travel to Pluto. Using the gravity and rotation of Jupiter, the probe will sling-shot itself towards Pluto. With an interactive program you can input the space craft’s speed, approach angle and planet’s speed. The computer then outputs the resulting speed after the swing- by

Importance of project To extend exploration outside our solar system Fuel Efficiency Position of planets at time of swing by Gravity Assist Objectives Exploration of Planets New Horizons Space Probe

Voyager I/II Exploration

Conservation of Energy and Momentum Kinetic Energy Energy of Motion –KE = ½ x Mass x Velocity 2 Momentum Quantity of Motion –Momentum = Mass x Velocity

How it works….. Spacecraft gains significant momentum and velocity while planet takes minute losses

Derivation of Velocity Equations V 2 = 2U + V 1 Velocity after 180 degree turn V2 = (v1 + 2u) [ 1 - 4uv1(1-cos(θ))/ (v1+2u)2 ]½ Velocity considering different angles of approach

Java code, written in applet form import java.awt.Graphics; import javax.swing.*; public class testaa extends JApplet { double v2; public void init() { String v2b,ss1b,aa1b,ps1b,thetab; double ss1,aa1,ps1,theta; ss1b = JOptionPane.showInputDialog("Spacecraft speeds typically vary between 0 and 100 Km/s\nPlease input your spacecraft speed: "); ss1 = Double.parseDouble(ss1b);

code will allows user to interact with world wide web. aa1b = JOptionPane.showInputDialog("Approach angle between 0 and 90 degrees\nPlease input your approach angle: "); aa1 = Double.parseDouble(aa1b); ps1b = JOptionPane.showInputDialog("Mercury: 47.89\nVenus: 35.03\nEarth:29.79\nMars: 24.13\nJupiter:13.06\nSaturn:9.64\nUranus: 6.81\nNeptune:5.43\nPluto:4.74\nPlease input your planet speed: "); ps1 = Double.parseDouble(ps1b);

Final math is fairly simple. theta=aa1*Math.PI/180; v2 = (ss1+2*ps1)*Math.sqrt(1-4*ps1*ss1*(1- Math.cos(theta))/((ss1+2*ps1)*(ss1+2*ps1))); } public void paint( Graphics g ) { super.paint( g ); g.drawRect(15,10,270,20); g.drawString("Final speed in Km/s is: " + v2, 25, 25 ); } }

Triangle equation method, compare results with different methods We know the orbital radius of Jupiter, X 1, initial velocity towards Jupiter, V 1, the orbital radius of Pluto, X 3 and the time we want to take for the mission, t. Using law of sines we can find the last side X 2, and X 1 +X 2 = d, then d/t = V a, average speed. Rewriting, V a = (V 1 +V 2 )/2 = d/t. Thus, V 2 = 2d/t – V 1

Compare results Using the same inputs for both the sling equation and the triangle equation we have a V 2 of 31.6 Km/s for the sling compared a V 2 of 31.4 Km/s for the triangle. Initial V 1, and radius of Jupiter X 1 Solved distance X 2, and unknown V 2  Radius of Pluto, X 3

Validation The triangle equation yields the same results as the slingshot equation, however the triangle equation requires more calculations and is very static, not very malleable for other applications The slingshot equation has much greater applications and is more amendable to parametric and vector calculations. Furthermore, the slingshot method fits directly into other celestial mechanic equations.