# Advanced Physics

From Math and Science

Advanced Physics with calculus and programming.

## Project 1: Gravitational motion

- Throw a ball into the air and map its height with time.
- Plot the time/height graph and find the best-fit line.
- Don't forget to record your R-squared value

- Based on your equation determine (using the calculus) the:
- gravitational acceleration
- maximum height of the ball
- initial velocity of the ball
- initial height of the ball

- Compare the actual gravitational acceleration on the surface of the Earth to the acceleration you calculated from your experiment.
- Use a graph to show how the curve you got from your data compares to the curve predicted using Earth's surface acceleration of 9.8 m/s^2.

- Determine the maximum height of the ball if thrown on Mars with the same initial velocity and initial height determined from your data.
- Plot a series of curves showing the maximum height on Earth, Mars, the Moon, Jupiter.
- Write a computer program that uses the physics of motion to show the motion of the ball.
- Write a full report (How to write a lab report) for this project.

## Project 2: Force and momemtum

- Build a spaceship (in vpython)
- Your ship must have the following properties:
- mass: maximum mass of the ship, which will be the sum of the reaction mass, weapon mass, and life support mass
- reaction_mass: the amount of mass you can use to maneuver
- ls_mass: life support mass. Must be at least 10% of the total mass
- weapon_mass: the mass that can be used for weapons.

- Your ship must have the following properties:
- Make your ship move by expelling the reaction msss.
- Starting with either conservation of momentum or Newton's second law (F=ma), write the equations that you will use to calculate the acceleration, velocity, and position of your ship. You will need to use the "in the moment" formulas.
- it may help to do the calculations for the ejection of one mass unit (say you shoot out a basketball or an elephant)

- Put the formulas into your program and make your ship achieve the following objectives. FOR ALL OF THESE OBJECTIVES YOUR STARTING POSITION WILL BE THE ORIGIN (0,0,0), AND YOUR STARTING VELOCITY WILL BE (0,0,0):
- Accelerate until you reach
**Marker 1. located 1000 meters away in the x-direction (at (1000,0,0))**.- Determine if your model works correctly by comparing the motion of your spaceship to the analytical solution.
- Restrict your acceleration to 10g's.

- Move to Marker 1, turn around, and return to your starting position.
- Come to a complete stop within 10m of the origin.
- Bonus: minimize the time it takes for the trip (you are still restricted to a maximum of 10g acceleration).

- Place two more markers:
**Marker 2. located at (1000,1000,0)**and**Marker 3. located at (0,1000,0)**. Fly around those markers and return to your starting position.- Bonus: pass within 20 m of each marker.

- Give your markers mass (a large mass compared to your spaceship) and adjust your model to include the force due to gravity.
- Show your mathematics

- Complete the first three objectives again, but this time be sure to avoid hitting your markers.
- Bonus: use the markers to slingshot your spaceship.

- Accelerate until you reach

- Starting with either conservation of momentum or Newton's second law (F=ma), write the equations that you will use to calculate the acceleration, velocity, and position of your ship. You will need to use the "in the moment" formulas.

### Project 2: Final Report

Show that your model of a spaceship works by comparing the time your ship takes to get to the first marker (at (1000,0,0)) if it constantly accelerates toward it (no turnover or deceleration), to the time calculated based on the acceleration imparted to your ship from the reaction mass. To simplify the math, we will assume that the amount of reaction mass used for this trip is small compared to the mass of the spaceship, so you can assume a constant acceleration (be sure to state this assumption clearly in your report.

- Be sure to plot a graph comparing the time taken by the model to the analytical solution.