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Step | Formula / Calculation | Reasoning |
---|---|---|

1 | [katex]v = 101 , \text{km/h}[/katex], [katex]t = 0.500 , \text{hr}[/katex] | Defining the constant speed and time for both boats. |

2 | [katex]v_{\text{m/s}} = \frac{101 \times 1000}{3600}[/katex] | Converting the speed from km/h to m/s. |

3 | [katex]v_{\text{west, boat 1}} = v_{\text{m/s}} \cos(\theta_{\text{boat 1}})[/katex], [katex]v_{\text{south, boat 1}} = v_{\text{m/s}} \sin(\theta_{\text{boat 1}})[/katex] | Decomposing Boat 1’s speed into westward and southward components. |

4 | [katex]v_{\text{west, boat 2}} = v_{\text{m/s}} \cos(\theta_{\text{boat 2}})[/katex], [katex]v_{\text{south, boat 2}} = v_{\text{m/s}} \sin(\theta_{\text{boat 2}})[/katex] | Decomposing Boat 2’s speed into westward and southward components. |

5 | [katex]\text{Distance west, boat 1} = v_{\text{west, boat 1}} \times t[/katex], [katex]\text{Distance south, boat 1} = v_{\text{south, boat 1}} \times t[/katex] | Calculating the westward and southward distances for Boat 1. |

6 | [katex]\text{Distance west, boat 2} = v_{\text{west, boat 2}} \times t[/katex], [katex]\text{Distance south, boat 2} = v_{\text{south, boat 2}} \times t[/katex] | Calculating the westward and southward distances for Boat 2. |

7 | [katex]\text{Difference west} = \text{Distance west, boat 1} – \text{Distance west, boat 2}[/katex] | Calculating the difference in the westward distance between the two boats. |

8 | [katex]\text{Difference south} = \text{Distance south, boat 2} – \text{Distance south, boat 1}[/katex] | Calculating the difference in the southward distance between the two boats. |

9 | [katex]\text{Difference west} \approx 0.98[/katex] km, [katex]\text{Difference south} \approx 1.61[/katex] km | Evaluating the differences in distances. |

During the half-hour journey, Boat 1 travels approximately 0.98 km farther west compared to Boat 2, and Boat 2 travels approximately 1.61 km farther south compared to Boat 1.

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- Statistics

Advanced

Mathematical

GQ

Gregory was walking through the halls of the school when he realized that he was walking in perpendicular directions and he could easily calculate his displacement using the incredibly useful techniques he learned in physics. He recognized that he walked 12.5 meters left and then 18.9 meters down. How far must he walk to the right so that his resultant displacement is 20.1 m?

- Vectors

Intermediate

Mathematical

FRQ

While Santa was delivering presents to the children of Nashville, Tennessee he experienced a strong wind perpendicular to his motion.

- Vectors

Intermediate

Mathematical

GQ

A person is standing at the edge of the water and looking out at the ocean. The height of the person’s eyes above the water is *h* = 1.8 m, and the radius of the Earth is *R* = 6.38 × 10^{6} m. How far is it to the horizon (in meters)? In other words, find the distance d from the person’s eyes to the horizon.

*(Note: At the horizon the angle between the line of sight and the radius of the earth is 90°)*

- Vectors

Intermediate

Mathematical

FRQ

A seagull first flies 160 m North, the heads up 120.65 m 18.43° North of West. After you land:

- Vectors

Intermediate

Mathematical

GQ

Determine the sum of the 3 vectors given below. Give the resultant (R) in terms of (a) vector components (b) resultant vector.

Vectors:

[katex] \vec{A} = 26.5 m [/katex] @ at 56° NW

[katex] \vec{B} = 44 m [/katex] @ at 28° NE

[katex] \vec{C} = 31 m [/katex] South

- Vectors

- .98 km
- 1.61 km

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Kinematics | Forces |
---|---|

[katex]\Delta x = v_i t + \frac{1}{2} at^2[/katex] | [katex]F = ma[/katex] |

[katex]v = v_i + at[/katex] | [katex]F_g = \frac{G m_1m_2}{r^2}[/katex] |

[katex]a = \frac{\Delta v}{\Delta t}[/katex] | [katex]f = \mu N[/katex] |

[katex]R = \frac{v_i^2 \sin(2\theta)}{g}[/katex] |

Circular Motion | Energy |
---|---|

[katex]F_c = \frac{mv^2}{r}[/katex] | [katex]KE = \frac{1}{2} mv^2[/katex] |

[katex]a_c = \frac{v^2}{r}[/katex] | [katex]PE = mgh[/katex] |

[katex]KE_i + PE_i = KE_f + PE_f[/katex] |

Momentum | Torque and Rotations |
---|---|

[katex]p = m v[/katex] | [katex]\tau = r \cdot F \cdot \sin(\theta)[/katex] |

[katex]J = \Delta p[/katex] | [katex]I = \sum mr^2[/katex] |

[katex]p_i = p_f[/katex] | [katex]L = I \cdot \omega[/katex] |

Simple Harmonic Motion |
---|

[katex]F = -k x[/katex] |

[katex]T = 2\pi \sqrt{\frac{l}{g}}[/katex] |

[katex]T = 2\pi \sqrt{\frac{m}{k}}[/katex] |

Constant | Description |
---|---|

[katex]g[/katex] | Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface |

[katex]G[/katex] | Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex] |

[katex]\mu_k[/katex] and [katex]\mu_s[/katex] | Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion. |

[katex]k[/katex] | Spring constant, in [katex]\text{N/m}[/katex] |

[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex] | Mass of the Earth |

[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex] | Mass of the Moon |

[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex] | Mass of the Sun |

Variable | SI Unit |
---|---|

[katex]s[/katex] (Displacement) | [katex]\text{meters (m)}[/katex] |

[katex]v[/katex] (Velocity) | [katex]\text{meters per second (m/s)}[/katex] |

[katex]a[/katex] (Acceleration) | [katex]\text{meters per second squared (m/s}^2\text{)}[/katex] |

[katex]t[/katex] (Time) | [katex]\text{seconds (s)}[/katex] |

[katex]m[/katex] (Mass) | [katex]\text{kilograms (kg)}[/katex] |

Variable | Derived SI Unit |
---|---|

[katex]F[/katex] (Force) | [katex]\text{newtons (N)}[/katex] |

[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) | [katex]\text{joules (J)}[/katex] |

[katex]P[/katex] (Power) | [katex]\text{watts (W)}[/katex] |

[katex]p[/katex] (Momentum) | [katex]\text{kilogram meters per second (kgm/s)}[/katex] |

[katex]\omega[/katex] (Angular Velocity) | [katex]\text{radians per second (rad/s)}[/katex] |

[katex]\tau[/katex] (Torque) | [katex]\text{newton meters (Nm)}[/katex] |

[katex]I[/katex] (Moment of Inertia) | [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex] |

[katex]f[/katex] (Frequency) | [katex]\text{hertz (Hz)}[/katex] |

General Metric Conversion Chart

Conversion Example

Example of using unit analysis: Convert 5 kilometers to millimeters.

Start with the given measurement:

`[katex]\text{5 km}[/katex]`

Use the conversion factors for kilometers to meters and meters to millimeters:

`[katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]`

Perform the multiplication:

`[katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]`

Simplify to get the final answer:

`[katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]`

Prefix | Symbol | Power of Ten | Equivalent |
---|---|---|---|

Pico- | p | [katex]10^{-12}[/katex] | 0.000000000001 |

Nano- | n | [katex]10^{-9}[/katex] | 0.000000001 |

Micro- | µ | [katex]10^{-6}[/katex] | 0.000001 |

Milli- | m | [katex]10^{-3}[/katex] | 0.001 |

Centi- | c | [katex]10^{-2}[/katex] | 0.01 |

Deci- | d | [katex]10^{-1}[/katex] | 0.1 |

(Base unit) | – | [katex]10^{0}[/katex] | 1 |

Deca- or Deka- | da | [katex]10^{1}[/katex] | 10 |

Hecto- | h | [katex]10^{2}[/katex] | 100 |

Kilo- | k | [katex]10^{3}[/katex] | 1,000 |

Mega- | M | [katex]10^{6}[/katex] | 1,000,000 |

Giga- | G | [katex]10^{9}[/katex] | 1,000,000,000 |

Tera- | T | [katex]10^{12}[/katex] | 1,000,000,000,000 |

- Some answers may be slightly off by 1% depending on rounding, etc.
- Answers will use different values of gravity. Some answers use 9.81 m/s
^{2}, and other 10 m/s^{2 }for calculations. - Variables are sometimes written differently from class to class. For example, sometime initial velocity [katex] v_i [/katex] is written as [katex] u [/katex]; sometimes [katex] \Delta x [/katex] is written as [katex] s [/katex].
- Bookmark questions that you can’t solve so you can come back to them later.
- Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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