A block of mass m is moving on a horizontal frictionless surface with a speed v_0 as it approaches a block of mass 2m which is at rest and has an ideal spring attached to one side.

When the two blocks collide, the spring is completely compressed and the two blocks momentarily move at the same speed, and then separate again, each continuing to move.

- (a) Briefly explain why the two blocks have the same speed when the spring is completely compressed.
*(3 points)* - (b) Determine the speed vf of the two blocks while the spring is completely compressed.
*(3 points)* - (c) Determine the kinetic energy of the two blocks as they move together with the same speed.
*(3 points)* - (d) When the spring expands, the blocks are again separated, and the spring returns its compressed potential energy to kinetic energy in the two blocks. On a graph, sketch a graph of kinetic energy vs. time from the time block m approaches block 2m until the two blocks are separated after the collision. *You can upload an image of this or describe the graph.
*(3 points)* - (e) Write the equations that could be used to solve for the speed of each block after they have separated. It is not necessary to solve these equations for the two speeds.
*(3 points)*

- When the spring is compressed, both blocks have 0 relative velocity, meaning they move at the same speed. This is similar where two objects move at the same speed after an inelastic collision.
- v_0/3
- \frac{1}{6} mv_0^2
- The graph of KE should be a horizontal line followed by a downwards curved dip, followed by a horizontal line at the original position on the KE axis.
- mv_0 = mv_{f1} + mv_{f2} \quad \text{and} \quad \frac{1}{2}mv_0^2 = \frac{1}{2}mv_{f1}^2 + \frac{1}{2}mv_{f2}^2

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- Statistics

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MCQ

Consider the following cases of inelastic collisions.

Case (1) – A car moving at 75 mph collides with another car of equal mass moving at 75 mph in the opposite direction and comes to a stop.

Case (2) A car moving at 75 mph hits a stationary steel wall and rolls backs.

The collision time is the same for both cases. In which of these cases would result in the greatest impact force?

- Momentum

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Find the escape speed from a planet of mass 6.89 x 10^{25} kg and radius 6.2 x 10^{6} m.

- Circular Motion, Energy

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A vehicle is moving at a speed of 12.3 m/s on a decline when the brakes of all four wheels are fully applied, causing them to lock. The slope of the decline forms an angle of 18.0 degrees with the horizontal plane. Given that the coefficient of kinetic friction between the tires and the road surface is 0.650.

- Energy, Linear Forces

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A big bird has a mass of about 0.021 kg. Suppose it does 0.36 J of work against gravity, so that it ascends straight up with a net acceleration of 0.625 m/s^{2}. How far up does it move?

- Energy

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A 84.4 kg climber is scaling the vertical wall. His safety rope is made of a material that behaves like a spring that has a spring constant of 1.34 x 10^{3} N/m. He accidentally slips and falls 0.627 m before the rope runs out of slack. How much is the rope stretched when it breaks his fall and momentarily brings him to rest?

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A pendulum consists of a ball of mass m suspended at the end of a massless cord of length L . The pendulum is drawn aside through an angle of 60° with the vertical and released. At the low point of its swing, the speed of the pendulum ball is

- Centripetal Acceleration, Circular Motion, Energy, Pendulums, Simple Harmonic Motion

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A kickball is rolled by the pitcher at a speed of 10 m/s and it is kicked by another student. The kickball deforms a little during the kick, and then rebounds with a velocity of 15 m/s as its shape restores to a perfect sphere. Select all that must be true about the kickball and the kicking foot system.

- Energy, Momentum

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A rocket explodes into two fragments, one 25 times heavier than the other. The change in momentum of the lighter fragment is

- Impulse, Momentum

- When the spring is compressed, both blocks have 0 relative velocity, meaning they move at the same speed. This is similar where two objects move at the same speed after an inelastic collision.
- v_0/3
- \frac{1}{6} mv_0^2
- The graph of KE should be a horizontal line followed by a downwards curved dip, followed by a horizontal line at the original position on the KE axis.
- mv_0 = mv_{f1} + mv_{f2} \quad \text{and} \quad \frac{1}{2}mv_0^2 = \frac{1}{2}mv_{f1}^2 + \frac{1}{2}mv_{f2}^2

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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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