# Part (a): Time Elapsed from Leaving the Table to Hitting the Floor
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | t = \sqrt{\frac{2h}{g}} | Since the motion in the y-direction is a free fall, we use the kinematic equation y = \frac{1}{2}gt^2 for the vertical motion. Solving for t gives the time it takes to fall a distance h . |
# Part (b): Horizontal Component of the Velocity of the Block Just Before It Hits the Floor
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | v_x = \frac{D}{t} | Using the result from part (a) and substituting t = \sqrt{\frac{2h}{g}} , we get v_x = \frac{D}{\sqrt{\frac{2h}{g}}} = \sqrt{\frac{D^2g}{2h}} . This is the velocity necessary to cover horizontal distance D in time t . |
| 2 | v_x = \frac{D}{\sqrt{\frac{2h}{g}}} | Replace t with the equation found in part a. |
# Part (c): Work Done on the Block by the Spring
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | El = KE | The work done by the spring (elastic energy) is transforms into kinetic energy. Since we found the velocity in the previous part we can solve for the kinetic energy. |
| 2 | El = KE = \frac{1}{2}mv^2 | Formula for kinetic energy |
| 3 | KE = \frac{1}{2} m \left(\frac{D}{\sqrt{\frac{2h}{g}}}\right)^2 | Substitute in velocity found from previous step. |
| 4 | KE = \frac{mgD^2}{4h} | Simplify equation. Note that the Kinetic energy is the work done by the spring. |
# Part (d): Spring Constant
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \frac{1}{2}kx^2 = \frac{1}{2}mv^2 | The spring energy (EL) is equal to the kinetic energy as mentioned in part c. Hence we can set EL = KE and solve for k. |
| 2 | k = \frac{mv^2}{x^2} | Solve for k |
| 3 | k = \frac{m \left(\frac{D}{\sqrt{\frac{2h}{g}}}\right)^2}{x^2} | Substitute in the v , to get the final equation in terms of M, x, D, h, |
| 4 | k = \frac{mD^2 g}{2hx^2} | Simplify |
These steps address each part of the query based on principles of mechanics, conservation of energy, and kinematic equations.
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On a distant planet, golf is just as popular as it is on earth. A golfer tees off and drives the ball 3.5 times as far as he would have on earth, given the same initial velocities on both planets. The ball is launched at a speed of 45 m/s at an angle of 29° above the horizontal. When the ball lands, it is at the same level as the tee. On the distant planet find:
A arrow is shot horizontally from a distance of 20 meters away. It lands .05 meters below the center of the target. If air resistance is negligible what was the initial speed of the arrow?
An airplane with a speed of 97.5 m/s is climbing upward at an angle of 50.0° with respect to the horizontal. When the plane’s altitude is 732 m, the pilot releases a package.
A 81 kg student dives off a 45 m tall bridge with an 18 m long bungee cord tied to his feet and to the bridge. You can consider the bungee cord to be a flexible spring. What spring constant must the bungee cord have for the student’s lowest point to be 2.0 m above the water?
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.
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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
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} | |
Nano- | n | 10^{-9} | |
Micro- | µ | 10^{-6} | |
Milli- | m | 10^{-3} | |
Centi- | c | 10^{-2} | |
Deci- | d | 10^{-1} | |
(Base unit) | – | 10^{0} | |
Deca- or Deka- | da | 10^{1} | |
Hecto- | h | 10^{2} | |
Kilo- | k | 10^{3} | |
Mega- | M | 10^{6} | |
Giga- | G | 10^{9} | |
Tera- | T | 10^{12} |
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