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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | [katex] PE_{\text{initial}} = 0 [/katex] | The object is initially on the ground, so its initial potential energy is zero (since potential energy is relative to height above a reference point). |
| 2 | [katex] PE_{\text{max}} = mgH [/katex] | At maximum height [katex] H [/katex], all the kinetic energy has been converted to potential energy. Potential energy at height [katex] H [/katex] is given by [katex] mgH [/katex], where [katex] m [/katex] is the mass, [katex] g [/katex] is the acceleration due to gravity, and [katex] H [/katex] is the height. |
| 3 | [katex] PE_{\text{half}} = mg \frac{H}{2} [/katex] | At half the maximum height [katex] \frac{H}{2} [/katex], the potential energy is [katex] mg \frac{H}{2} [/katex]. |
| 4 | [katex] KE_{\text{initial}} = mgH [/katex] | Using the principle of conservation of mechanical energy, the initial kinetic energy must equal the potential energy at maximum height, since there is no kinetic energy at that point (the object is momentarily stationary). |
| 5 | [katex] KE_{\text{half}} = KE_{\text{initial}} – PE_{\text{half}} [/katex] | To find the kinetic energy at height [katex] \frac{H}{2} [/katex], subtract the potential energy at that height from the total initial mechanical energy. |
| 6 | [katex] KE_{\text{half}} = mgH – mg \frac{H}{2} = mg \frac{H}{2} [/katex] | Simplifying, the kinetic energy at [katex] \frac{H}{2} [/katex] is [katex] mg \frac{H}{2} [/katex], or half the initial kinetic energy. |
| 7 | [katex] ME_{\text{initial}} = ME_{\text{half}} = KE_{\text{initial}} [/katex] | Total mechanical energy is conserved (since air resistance is negligible). The mechanical energy at any point during the motion is the sum of potential and kinetic energies and is equal to the initial total mechanical energy. |
| 8 | [katex](a)[/katex] False, [katex](b)[/katex] False, [katex](c)[/katex] False, [katex](d)[/katex] True, [katex](e)[/katex] False | Evaluating each statement: (a) Potential energy at [katex] \frac{H}{2} [/katex] is not half of the initial (it’s [katex] \frac{1}{2} mgH [/katex]). (b) Speed is not directly proportional in this manner. (c) Total mechanical energy is constant, not halved. (d) Kinetic energy at [katex] \frac{H}{2} [/katex] is indeed half of the initial kinetic energy, as shown in the calculations. (e) Since (d) is true, (e) is false. |
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A snowboarder starts from rest and slides down a \(32^\circ\) incline that’s \(75 \, \text{m}\) long.
An object is moving in the \( +x \) direction and begins to slow down. What must be true about its acceleration?
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A boat is rowed directly upriver at a speed of \(2.5 \, \text{m/s}\) relative to the water. Viewers on the shore find that it is moving at only \(0.5 \, \text{m/s}\) relative to the shore. What is the speed of the river? Is it moving with or against the boat?
In which of these cases is the rate of change of the particle’s displacement constant?
A person is making homemade ice cream. She exerts a force of magnitude \(23 \, \text{N}\) on the free end of the crank handle on the ice-cream maker, and this end moves on a circular path of radius \(0.25 \, \text{m}\). The force is always applied parallel to the motion of the handle. If the handle is turned once every \(1.7 \, \text{s}\), what is the average power being expended?
A car travels at \( 20 \, \text{m/s} \) for \( 5 \, \text{mins} \) and then travels another \( 2 \, \text{km} \) at \( 40 \, \text{m/s} \). What is the total distance traveled and time of travel for the car?
An airplane is traveling \( 900. \) \( \text{km/h} \) in a direction \( 38.5^\circ \) west of north.
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.
A \( 1.0 \)\( \text{-kg} \) object is moving with a velocity of \( 6.0 \) \( \text{m/s} \) to the right. It collides and sticks to a \( 2.0 \)\( \text{-kg} \) object moving with a velocity of \( 3.0 \) \( \text{m/s} \) in the same direction. How much kinetic energy was lost in the collision?
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| Kinematics | Forces |
|---|---|
| \(\Delta x = v_i t + \frac{1}{2} at^2\) | \(F = ma\) |
| \(v = v_i + at\) | \(F_g = \frac{G m_1 m_2}{r^2}\) |
| \(v^2 = v_i^2 + 2a \Delta x\) | \(f = \mu N\) |
| \(\Delta x = \frac{v_i + v}{2} t\) | \(F_s =-kx\) |
| \(v^2 = v_f^2 \,-\, 2a \Delta x\) |
| Circular Motion | Energy |
|---|---|
| \(F_c = \frac{mv^2}{r}\) | \(KE = \frac{1}{2} mv^2\) |
| \(a_c = \frac{v^2}{r}\) | \(PE = mgh\) |
| \(T = 2\pi \sqrt{\frac{r}{g}}\) | \(KE_i + PE_i = KE_f + PE_f\) |
| \(W = Fd \cos\theta\) |
| Momentum | Torque and Rotations |
|---|---|
| \(p = mv\) | \(\tau = r \cdot F \cdot \sin(\theta)\) |
| \(J = \Delta p\) | \(I = \sum mr^2\) |
| \(p_i = p_f\) | \(L = I \cdot \omega\) |
| Simple Harmonic Motion | Fluids |
|---|---|
| \(F = -kx\) | \(P = \frac{F}{A}\) |
| \(T = 2\pi \sqrt{\frac{l}{g}}\) | \(P_{\text{total}} = P_{\text{atm}} + \rho gh\) |
| \(T = 2\pi \sqrt{\frac{m}{k}}\) | \(Q = Av\) |
| \(x(t) = A \cos(\omega t + \phi)\) | \(F_b = \rho V g\) |
| \(a = -\omega^2 x\) | \(A_1v_1 = A_2v_2\) |
| 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] |
Metric Prefixes
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] | |
Nano- | n | [katex]10^{-9}[/katex] | |
Micro- | µ | [katex]10^{-6}[/katex] | |
Milli- | m | [katex]10^{-3}[/katex] | |
Centi- | c | [katex]10^{-2}[/katex] | |
Deci- | d | [katex]10^{-1}[/katex] | |
(Base unit) | – | [katex]10^{0}[/katex] | |
Deca- or Deka- | da | [katex]10^{1}[/katex] | |
Hecto- | h | [katex]10^{2}[/katex] | |
Kilo- | k | [katex]10^{3}[/katex] | |
Mega- | M | [katex]10^{6}[/katex] | |
Giga- | G | [katex]10^{9}[/katex] | |
Tera- | T | [katex]10^{12}[/katex] |
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