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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[\text{Let } v_x \text{ be the constant upward speed reached after the acceleration phase.}\] | Define \(v_x\) because the exact numerical value of the elevator’s cruising speed is not given but is required to express the graph algebraically. |
| 2 | \[a = \frac{v_x}{2}\] | For the first \(2\,\text{s}\) the elevator starts from rest and reaches \(v_x\). Using \(v_x = v_i + a t\) with \(v_i = 0\) and \(t = 2\,\text{s}\) gives \(a = v_x/2\). |
| 3 | \[v(t)=\frac{v_x}{2}t,\; 0 \le t \le 2\] | During the acceleration interval the velocity increases linearly from \(0\) to \(v_x\) with slope \(a = v_x/2\). |
| 4 | \[v(t)=v_x,\; 2 < t \le 7\] | The elevator travels at its steady speed for the next \(5\,\text{s}\) (from \(t=2\) to \(t=7\) seconds), producing a horizontal line on the graph. |
| 5 | \[a_d = -\frac{v_x}{3}\] | To come gently to rest in the final \(3\,\text{s}\), the required (constant) deceleration is \(a_d = \frac{0 – v_x}{3}\). |
| 6 | \[v(t)=v_x\left(1-\frac{t-7}{3}\right),\; 7 < t \le 10\] | Starting from \(v_x\) at \(t=7\,\text{s}\) and decreasing linearly to \(0\) at \(t=10\,\text{s}\) with slope \(a_d\). |
| 7 | \[v(t)=\begin{cases}\tfrac{v_x}{2}t & 0 \le t \le 2 \\ v_x & 2 < t \le 7 \\ v_x\!\left(1-\tfrac{t-7}{3}\right) & 7 < t \le 10 \end{cases}\] | This piece-wise function captures the entire velocity–time graph: a straight line up, a plateau, and a straight line down to zero. |
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Which graph below shows that one of the runners started 10 meters further ahead of the other? Assume the y-axis is measured in meters and the x-axis is measured in seconds.
The graph above shows velocity as a function of time for an object moving along a straight line. For which of the following sections of the graph is the acceleration constant and nonzero?
A rollercoaster leaves the station at rest. Its speed increases steadily for \( 6 \) \( \text{s} \) as it heads down the first drop. The ride then levels out and it moves at a constant speed for \( 4 \) \( \text{s} \) before hitting the brakes and stopping in \( 3 \) \( \text{s} \). Draw the velocity vs. time graph or explain it in terms of functions.
A cart begins to move from rest on a horizontal track. Which of the following correctly indicates the magnitude of the average velocity of the cart during the interval shown and provides a valid explanation?
Hint: when solving this, its consider that the area of the acceleration vs time graph tells you the change in velocity.
In which of these cases is the rate of change of the particle’s displacement constant?
Which of the following graphs shows runners moving at the same speed? Assume the \(y\)-axis is measured in meters and the \(x\)-axis is measured in seconds.
Which statement is true about the distances the two objects have traveled at time \( t_f \)?
An object’s velocity \(v\) as a function of time \(t\) is given in the graph. Which of the following statements is true about the motion of the object?
The displacement \(x\) of an object moving in one dimension is shown above as a function of time \(t\). The velocity of this object must be
Which of the following graphs represent an object at rest?
\[v(t)=\begin{cases}\tfrac{v_x}{2}t & 0 \le t \le 2 \\ v_x & 2 < t \le 7 \\ v_x\!\left(1-\tfrac{t-7}{3}\right) & 7 < t \le 10 \end{cases}\]
You can also view the drawn out graph here.
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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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