Trajectories

By: Zack Stutts

Freddy Sonnenwald

Uses Cold War

– Ballistic Missile• Nuclear• Biological• Conventional

Napoleonic War– Cannon

Sports/Entertainments– Basketball– Football– Dare Devil Stunts

The Question

• Velocity• Angle• Time steps• Gravity

Operations

• Validation– Research

• Compare Models

• Verification– Advanced Calculations

• Euler’s Method

• An Excel Workbook• A Java Program

Architecture

• Microsoft Excel• Sun Java

Estimating Math

• Euler’s Method: New = Old + Change– Vy new = Vy old + g · Δt– Vx new = Vx old + ø– X new = X old + Vx · Δt– Y new = Y old + Vy · Δt

Exact Math

• Exact Mathematical Equation– Y at t = Y init + Vy init · t – (1/2 · g · t²)– X at t = X init + Vx init · t

Excel Estimated• Euler’s Method

– Difference < 15 m• Reasonable Distance

– Fit’s on GraphDropping Estimated

0

2000

4000

6000

8000

10000

0 5000 10000 15000 20000 25000X- Distance Estimated

Excel Exact

• Exact Calculations• Reasonable Distance

– Fits on GraphDropping Exact

0

2000

4000

6000

8000

10000

0 5000 10000 15000 20000 25000X- Distance Exact

Java

• Both exact/estimated• Finds height/distance

Results

• Euler’s Method Perfect• Time steps dependant• Velocity effective• 45 degrees high range• Gravity Important

Sources

• http://www.physicsclassroom.com/Class/vectors/U3L2e.html the formulas

• http://www.civilwarartillery.com/projectiles/spherical/IIIA3.htm velocity of a cannonball

• http://id.mind.net/~zona/mstm/physics/mechanics/curvedMotion/projectileMotion/generalSolution/generalSolution.html general information