# (a) Displacement during the first two seconds

The displacement is the area under the velocity-time graph from t = 0 to t = 2 seconds.

Step | Derivation/Formula | Reasoning |
---|---|---|

1 | \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} | The graph from t = 0 to t = 2 forms a right triangle. Calculate its area. |

2 | \text{Area} = \frac{1}{2} \times 2 \, \text{s} \times 4 \, \text{m/s} | Substitute the base (time interval) and height (velocity) into the area formula. |

3 | \text{Displacement} = 4 \,\text{meters} |
The area (in square units) represents the player’s displacement in meters. |

# (b) Displacement between t = 4 \, \text{s} and t = 9 \, \text{s}

Calculate the area under the velocity-time graph between t = 4 \, \text{s} and t = 9 \, \text{s}.

Step | Derivation/Formula | Reasoning |
---|---|---|

1 | A_{\text{rectangle}} = \text{base} \times \text{height} | Calculate the area of the rectangle from t = 4 to t = 8. |

2 | A_{\text{rectangle}} = (8 – 4) \, \text{s} \times 2 \, \text{m/s} = 8 \, \text{meters} | Substitute the base (4 seconds) and height (2 m/s) into the equation. |

3 | A_{\text{triangle}} = \frac{1}{2} \times (9 – 8) \, \text{s} \times 2 \, \text{m/s} = 1 \, \text{meter} | Calculate the area of the triangle from t = 8 to t = 9. |

4 | \text{Total displacement} = 8 \, \text{meters} + 1 \, \text{meter} = 9 \, \text{meters} | Sum of the rectangle and triangle areas give the total displacement. |

# (c) Displacement between t = 4 \, \text{s} and t = 10 \, \text{s}

Calculate the area under the velocity-time graph between t = 4 \, \text{s} and t = 10 \, \text{s}, including the negative area.

Step | Derivation/Formula | Reasoning |
---|---|---|

1 | A_{\text{rectangle}} = \text{base} \times \text{height} = 8 \, \text{meters} | Area calculated previously for rectangle from t = 4 to t = 8. |

2 | A_{\text{triangle1}} = \frac{1}{2} \times (9 – 8) \, \text{s} \times 2 \, \text{m/s} = 1 \, \text{meter} | Area calculated previously for triangle from t = 8 to t = 9. |

3 | A_{\text{triangle2}} = \frac{1}{2} \times (10 – 9) \, \text{s} \times (-2) \, \text{m/s} = -1 \, \text{meter} | Calculate the area (negative) for the triangle from t = 9 to t = 10. |

4 | \text{Total displacement} = 8 \, \text{meters} + 1 \, \text{meter} – 1 \, \text{meter} = 8 \, \text{meters} | Sum the areas of the rectangle and the two triangles. |

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- Statistics

Intermediate

Mathematical

MCQ

- Motion Graphs

Beginner

Conceptual

MCQ

The displacement x of an object moving in one dimension is shown above as a function of time t. The acceleration of this object must be

- 1D Kinematics, Motion Graphs

Intermediate

Conceptual

MCQ

Which pair of graphs represents the same 1- dimensional motion?

- 1D Kinematics, Motion Graphs

Beginner

Conceptual

MCQ

Which of the following graphs represent an object at rest?

- Motion Graphs

Intermediate

Conceptual

MCQ

Which of the following graphs represent an object having zero acceleration? (There could be more than one answer)

- Motion Graphs

(a) 4 m

(b) 9 m

(c) 8 m

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

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

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

a = \frac{\Delta v}{\Delta t} | f = \mu N |

R = \frac{v_i^2 \sin(2\theta)}{g} |

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

F_c = \frac{mv^2}{r} | KE = \frac{1}{2} mv^2 |

a_c = \frac{v^2}{r} | PE = mgh |

KE_i + PE_i = KE_f + PE_f |

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

p = m v | \tau = r \cdot F \cdot \sin(\theta) |

J = \Delta p | I = \sum mr^2 |

p_i = p_f | L = I \cdot \omega |

Simple Harmonic Motion |
---|

F = -k x |

T = 2\pi \sqrt{\frac{l}{g}} |

T = 2\pi \sqrt{\frac{m}{k}} |

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

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

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

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

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

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

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

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

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

s (Displacement) | \text{meters (m)} |

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

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

t (Time) | \text{seconds (s)} |

m (Mass) | \text{kilograms (kg)} |

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

F (Force) | \text{newtons (N)} |

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

P (Power) | \text{watts (W)} |

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

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

\tau (Torque) | \text{newton meters (Nm)} |

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

f (Frequency) | \text{hertz (Hz)} |

General Metric Conversion Chart

Conversion Example

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

Start with the given measurement:

`\text{5 km}`

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

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

Perform the multiplication:

`\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}`

Simplify to get the final answer:

`\boxed{5 \times 10^6 \, \text{mm}}`

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

Pico- | p | 10^{-12} | 0.000000000001 |

Nano- | n | 10^{-9} | 0.000000001 |

Micro- | µ | 10^{-6} | 0.000001 |

Milli- | m | 10^{-3} | 0.001 |

Centi- | c | 10^{-2} | 0.01 |

Deci- | d | 10^{-1} | 0.1 |

(Base unit) | – | 10^{0} | 1 |

Deca- or Deka- | da | 10^{1} | 10 |

Hecto- | h | 10^{2} | 100 |

Kilo- | k | 10^{3} | 1,000 |

Mega- | M | 10^{6} | 1,000,000 |

Giga- | G | 10^{9} | 1,000,000,000 |

Tera- | T | 10^{12} | 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 v_i is written as u ; sometimes \Delta x is written as s .
- 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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