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Explanation

- Statistics

*v* versus time *t* for an object in linear motion. Which of the following is a possible graph of position *x* versus time *t* for this object?

^{2} N is being pulled up vertically by a rope from the bottom of a cave that is 35.2 m deep. The maximum tension that the rope can withstand without breaking is 592 N. What is the shortest time, starting from rest, in which the person can be brought out of the cave?

*v*, and reaches a maximum height *h*. At what height was the baseball moving with one-half its original velocity? Assume air resistance is negligible.

You stand at the edge of a vertical cliff and throws a stone vertically upwards. The stone leaves your hand with a speed v = 8.0 m/s. The time between the stone leaving your hand and hitting the sea is 3.0 s. Assume air resistance is negligible. Calculate:

- The maximum height reached by the stone (the distance above the cliff)
- The time taken by the stone to reach its maximum height.
- The height of the cliff.

A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s.

- For how much time and what distance does the student have to run at 5.0 m/s before she overtakes the bus
- When she reaches the bus, how fast is the bus traveling?
- Sketch an x-t graph for both the student and the bus. Take x = 0 at the initial position

of the student. - The equations you used in part (a) to find the time have a second solution, corresponding to a later time for which the student and the bus are again at the same place if they continue their specified motions. Explain the significance of this second solution. How fast is the bus traveling at this point?
- If the student’s top speed is 3.5 m/s, will she catch the bus?
- What is the minimum speed the student must have to just catch up with the bus? For what time and distance does she have to run that case?

A rocket is sent to shoot down an invading spacecraft that is hovering at an altitude of 1500 meters. The rocket is launched with an initial velocity of 180 m/s. Find the following:

- The minimum velocity needed to hit the spaceship.
- The time taken for the rocket to reach its maximum height when launched at 180 m/s.
- The maximum height the rocket reaches when launched at 180 m/s.
- The velocity and time of impact (assuming that the rocket collides with the spacecraft).

A whiffle ball is tossed straight up, reaches a highest point, and falls back down. Air resistance is *not* negligible. Which of the following statements are true?

- The ball’s speed is zero at the highest point.
- The ball’s acceleration is zero at the highest point.
- The ball takes a longer time to travel up to the highest point than to fall back down.

Can an object’s average velocity equal zero when object’s speed is greater than zero?

A rocket, initially at rest, is fired vertically upward with an acceleration of 12.0 m/s^{2}. At an altitude of 1.00 km, the rocket engine cuts off. Drag is negligible.

- How fast is the rocket traveling when the engine cuts off?
- What maximum height relative to the ground does the rocket reach before it begins falling back toward the earth?
- After free-falling, what is the rocket’s velocity just before it hits the earth?
- For what total amount of time was the rocket in the air (from initial launch to return to earth)?

^{2}. How far does the car go before it stops?

*x* direction. The cart has a constant acceleration in the +*x* direction with magnitude 3 m/s^{2} < a < 6 m/s^{2}?. Which of the following gives the possible range of the position of the cart at t = 1 s?

Two students are on a balcony 19.6 m above the street. One student throws a ball vertically downward at 14.7 m/s. At the same instant, the other student throws a ball vertically upward at the same speed. The second ball just misses the balcony on the way down.

- What is the difference in the time the balls spend in the air?
- What is the velocity of each ball as it strikes the ground?
- How far apart are the balls 0.800 s after they are thrown?

^{2}. At *t _{1}* the rocket engine is shut down and the sled moves with constant velocity

^{2}. What is the maximum reaction time allowed if the ranger is to avoid hitting the deer?

A coin is dropped from a hot air-balloon that is 250 m above the ground rising at 11 m/s upwards. For the coin, find the following:

- The maximum height reached (in meters)
- The position, 4.20 seconds after being released. (Assume up is positive)
- The velocity 4.20 seconds after being released. (Assume up is positive)
- The time ( in seconds) before it hits the ground

Priscilla the Penguin stands at the edge of a rock ledge and tosses a small ice cube directly upward with an initial velocity of *v _{o}*. The ice cube’s initial height above the ground is 3.25 m, and reaches it maximum height above the ground 0.586 s after being thrown. The ice cube then plummets to the ground, missing the edge of the rock ledge on its way down.

- Calculate the initial speed
*v*of the ice cube._{o} - Calculate the maximum height above the ground that the ice cube reaches.
- Calculate the amount of time it takes the ice cube to reach the ground after Pricilla throws it.
- Calculate the speed of the ice cube when it reaches the ground
- Calculate the height of the ice cube above the ground when it is traveling at 7.00 m/s in the downward direction.

Which of the following statements about the acceleration due to gravity is TRUE?

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

\Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2 | F = m \cdot a |

v = v_i + a \cdot t | F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} |

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

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

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

F_c = \frac{m \cdot v^2}{r} | KE = \frac{1}{2} m \cdot v^2 |

a_c = \frac{v^2}{r} | PE = m \cdot g \cdot h |

KE_i + PE_i = KE_f + PE_f |

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

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

J = \Delta p | I = \sum m \cdot r^2 |

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

Simple Harmonic Motion |
---|

F = -k \cdot 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} |

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 (kg·m/s)} |

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

\tau (Torque) | \text{newton meters (N·m)} |

I (Moment of Inertia) | \text{kilogram meter squared (kg·m}^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!