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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[A_x=-A\cos56^\circ\] | Vector \(\vec A\) points \(56^\circ\) north of west, so its horizontal component is westward (negative). |
| 2 | \[A_y=A\sin56^\circ\] | The vertical component of \(\vec A\) is northward (positive). |
| 3 | \[A_x\;{=}\;-14.8\,\text{m},\;\;A_y\;{=}\;22.0\,\text{m}\] | Substituted \(A=26.5\,\text{m}\) and the trigonometric values \(\cos56^\circ=0.559\) and \(\sin56^\circ=0.829\). |
| 4 | \[B_x=B\cos28^\circ,\;\;B_y=B\sin28^\circ\] | \(\vec B\) is \(28^\circ\) north of east, giving positive \(x\) and \(y\) components. |
| 5 | \[B_x\;{=}\;38.9\,\text{m},\;\;B_y\;{=}\;20.7\,\text{m}\] | Inserted \(B=44\,\text{m}\) with \(\cos28^\circ=0.883\) and \(\sin28^\circ=0.469\). |
| 6 | \[C_x=0,\;\;C_y=-31\,\text{m}\] | \(\vec C\) points straight south, so only a negative \(y\) component exists. |
| 7 | \[R_x=A_x+B_x+C_x\] | Horizontal components add algebraically. |
| 8 | \[R_x=-14.8+38.9+0=24.0\,\text{m}\] | Computed the net \(x\)-component. |
| 9 | \[R_y=A_y+B_y+C_y\] | Vertical components add algebraically. |
| 10 | \[R_y=22.0+20.7-31=11.6\,\text{m}\] | Computed the net \(y\)-component. |
| 11 | \[\boxed{\vec R = 24.0\,\hat\imath + 11.6\,\hat j\;\text{m}}\] | Expressed \(\vec R\) in component form. |
| 12 | \[\boxed{R_x = 24,\; R_y = 11.6}\] | Alternative way of writing final answer |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[R=\sqrt{R_x^2+R_y^2}\] | Pythagorean theorem gives the magnitude of the resultant in 2-D. |
| 2 | \[R=\sqrt{(24.0)^2+(11.6)^2}=26.7\,\text{m}\] | Inserted the component values from part (a). |
| 3 | \[\theta=\tan^{-1}\!\left(\frac{R_y}{R_x}\right)\] | Angle measured from the positive \(x\)-axis toward the positive \(y\)-axis. |
| 4 | \[\theta=\tan^{-1}\!\left(\frac{11.6}{24.0}\right)=25.8^\circ\] | Computed the arctangent; both components are positive, placing \(\vec R\) in the NE quadrant. |
| 5 | \[\boxed{R=26.7\,\text{m}\;@\;25.8^\circ\;\text{NE}}\] | Final magnitude and compass direction (degrees north of east). |
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A skier is accelerating down a \( 30.0^{\circ} \) hill at \( 3.80 \) \( \text{m/s}^2 \).
Vector \( V_1 \) is \( 6.0 \) units long and points along the negative \( y \) axis. Vector \( V_2 \) is \( 4.5 \) units long and points at \( +45^\circ \) to the positive \( x \) axis.
What does displacement mean in the context of motion?
You are piloting a small plane, and you want to reach an airport \(450 \, \text{km}\) due south in \(3.0 \,\text{hours}\). A wind is blowing from the west at \(50.0 \,\text{km/h}\). What heading and airspeed should you choose to reach your destination in time?
Vector \( A \) is \( 44.0 \) units and \( 28.0^\circ \) above the \( +x \) axis, vector \( B \) is \( 26.5 \) units and \( 56.0^\circ \) above the \( -x \) axis, and vector \( C \) is \( 31.0 \) units along the \( -y \) axis. Determine the resultant (sum) of the three vectors.
A seagull first flies \( 160 \, \text{m} \) North, then heads \( 120.65 \, \text{m} \) at \( 18.43^\circ \) North of West. After it lands:
Gregory was walking through the halls of the school when he realized that he was walking in perpendicular directions and he could easily calculate his displacement using the incredibly useful techniques he learned in physics. He recognized that he walked \(12.5\ \text{m}\) left and then \(18.9\ \text{m}\) down. How far must he walk to the right so that his resultant displacement is \(20.1\ \text{m}\)?
An airplane is traveling \( 900. \) \( \text{km/h} \) in a direction \( 38.5^\circ \) west of north.
An object is moving to the west at a constant speed. Three forces are exerted on the object. One force is \( 10 \) \( \text{N} \) directed due north, and another is \( 10 \) \( \text{N} \) directed due west. What is the magnitude and direction of the third force if the object is to continue moving to the west at a constant speed?
While Santa was delivering presents to the children of Nashville, Tennessee he experienced a strong wind perpendicular to his motion.
\(24.0\,\hat\imath+11.6\,\hat j\) or \(\boxed{R_x = 24,\; R_y = 11.6}\)
\(26.7\,\text{m}\text{ at }25.8^\circ\text{ NE}\)
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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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