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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( \Delta x = v_i t + \frac{1}{2} a t^2 \) | Use the equation for displacement under constant acceleration, substituting variables for given values. |
| 2 | \( v_i = -3 \, \text{m/s} \) | The initial velocity of the cart is given as \( 3 \, \text{m/s} \) in the \( -x \) direction. |
| 3 | \( t = 1 \, \text{s} \) | The time at which we want to find the position is \( 1 \, \text{s} \). |
| 4 | \( \Delta x = (-3 \, \text{m/s})(1 \, \text{s}) + \frac{1}{2} a (1 \, \text{s})^2 \) | Substitute \( v_i \) and \( t \) into the displacement equation. |
| 5 | \( \Delta x = -3 \, \text{m} + \frac{a}{2} \, \text{m}^2 \) | Simplify the equation. |
| 6 | \( 3 \, \text{m/s}^2 < a < 6 \, \text{m/s}^2 \) | The acceleration is given to be within this range. |
| 7 | \( \Delta x_{\text{min}} = -3 \, \text{m} + \frac{3 \, \text{m/s}^2}{2} \) | Calculate the displacement for the minimum acceleration \( a = 3 \, \text{m/s}^2 \). |
| 8 | \( \Delta x_{\text{min}} = -3 \, \text{m} + 1.5 \, \text{m} = -1.5 \, \text{m} \) | Simplify the equation for the minimum displacement. |
| 9 | \( \Delta x_{\text{max}} = -3 \, \text{m} + \frac{6 \, \text{m/s}^2}{2} \) | Calculate the displacement for the maximum acceleration \( a = 6 \, \text{m/s}^2 \). |
| 10 | \( \Delta x_{\text{max}} = -3 \, \text{m} + 3 \, \text{m} = 0 \, \text{m} \) | Simplify the equation for the maximum displacement. |
| 11 | \( x = x_0 + \Delta x \) | Use the initial position \( x_0 \) to find the final position. |
| 12 | \( x_{\text{min}} = 10 \, \text{m} – 1.5 \, \text{m} = 8.5 \, \text{m} \) | Calculate the final position for the minimum displacement. |
| 13 | \( x_{\text{max}} = 10 \, \text{m} + 0 \, \text{m} = 10 \, \text{m} \) | Calculate the final position for the maximum displacement. |
| 14 | \( 8.5 \, \text{m} < x < 10 \, \text{m} \) | Final range of the position is from \( 8.5 \, \text{m} \) to \( 10 \, \text{m} \), therefore the correct answer is option (b). |
Just ask: "Help me solve this problem."
A disk is initially rotating counterclockwise around a fixed axis with angular speed \( \omega_0 \). At time \( t = 0 \), the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?
Ball A is dropped from the top of a cliff. At the same time, Ball B is thrown straight upward from the ground at \( 30 \) \( \text{m/s} \). The two balls pass each other after \( 2.0 \) \( \text{s} \).
A body moving in the positive \( x \) direction passes the origin at time \( t = 0 \). Between \( t = 0 \) and \( t = 1 \, \text{second} \), the body has a constant speed of \( 24 \, \text{m/s} \). At \( t = 1 \, \text{second} \), the body is given a constant acceleration of \( 6 \, \text{m/s}^2 \) in the negative \( x \) direction. The position \( x \) of the body at \( t = 11 \, \text{seconds} \) is
A baseball is tossed from street level by a student straight up at a speed of \(25.3 \text{ m/s}\). After reaching maximum height, it is caught by another student on the roof of a building, \(17.4 \text{ m}\) above the street. How long did this take?
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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] |
General Metric Conversion Chart
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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