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| Step | Derivation/Formula | Explanation |
|---|---|---|
| 1 | \( y = y_0 + v_i t + \frac{1}{2} a t^2 \) | Use the kinematic equation for displacement. |
| 2 | Set \( y = 0 \), \( y_0 = 19.6 \) m, \( a = -9.8 \) m/s². | Define the variables. |
| 3 | For Ball A (downward): \( v_i = -14.7 \) m/s | Initial velocity downward is negative. |
| 4 | \( 0 = 19.6 -14.7 t – 4.9 t^2 \) | Substitute values into the equation. |
| 5 | \( 4.9 t^2 +14.7 t -19.6 = 0 \) | Rearrange into standard quadratic form. |
| 6 | Solve for \( t \): \( t = 1 \) s | Find the positive root of the quadratic equation. |
| 7 | For Ball B (upward): \( v_i = +14.7 \) m/s | Initial velocity upward is positive. |
| 8 | \( 0 = 19.6 +14.7 t – 4.9 t^2 \) | Substitute values into the equation. |
| 9 | \( -4.9 t^2 +14.7 t +19.6 = 0 \) | Simplify equation. |
| 10 | \( 4.9 t^2 -14.7 t -19.6 = 0 \) | Multiply both sides by -1. |
| 11 | Solve for \( t \): \( t = 4 \) s | Find the positive root of the quadratic equation. |
| 12 | \( \Delta t = t_{\text{Ball B}} – t_{\text{Ball A}} = 4 \, \text{s} – 1 \, \text{s} = 3 \, \text{s} \) | Calculate the difference in time. |
Answer: The difference in time the balls spend in the air is 3 seconds.
| Step | Derivation/Formula | Explanation |
|---|---|---|
| 1 | \( v = v_i + a t \) | Use the kinematic equation for velocity. |
| 2 | For Ball A: \( v = -14.7 \, \text{m/s} + (-9.8 \, \text{m/s}^2)(1 \, \text{s}) = -24.5 \, \text{m/s} \) | Compute final velocity for Ball A. |
| 3 | For Ball B: \( v = +14.7 \, \text{m/s} + (-9.8 \, \text{m/s}^2)(4 \, \text{s}) = -24.5 \, \text{m/s} \) | Compute final velocity for Ball B. |
Answer: Each ball strikes the ground with a velocity of -24.5 m/s downward.
| Step | Derivation/Formula | Explanation |
|---|---|---|
| 1 | \( y = y_0 + v_i t + \frac{1}{2} a t^2 \) | Use the kinematic equation for position. |
| 2 | For Ball A: \( y_{\text{A}} = 19.6 + (-14.7)(0.8) + \frac{1}{2}(-9.8)(0.8)^2 \) | Compute position of Ball A at \( t = 0.8 \) s. |
| 3 | \( y_{\text{A}} = 19.6 -11.76 -3.136 = 4.704 \, \text{m} \) | Simplify to find \( y_{\text{A}} \). |
| 4 | For Ball B: \( y_{\text{B}} = 19.6 + (+14.7)(0.8) + \frac{1}{2}(-9.8)(0.8)^2 \) | Compute position of Ball B at \( t = 0.8 \) s. |
| 5 | \( y_{\text{B}} = 19.6 +11.76 -3.136 = 28.224 \, \text{m} \) | Simplify to find \( y_{\text{B}} \). |
| 6 | \( \Delta y = y_{\text{B}} – y_{\text{A}} = 28.224 – 4.704 = 23.52 \, \text{m} \) | Calculate the distance between the balls. |
Answer: The balls are 23.52 meters apart 0.800 seconds after they are thrown.
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You are a bungee jumping fanatic and want to be the first bungee jumper on Jupiter. The length of your bungee cord is \( 45.0 \) \( \text{m} \). Free fall acceleration on Jupiter is \( 23.1 \) \( \text{m/s}^2 \). What is the ratio of your speed on Jupiter to your speed on Earth when you have dropped \( 45 \) \( \text{m} \)? Ignore the effects of air resistance and assume that you start at rest.
A car starts from rest and accelerates uniformly over a time of 5 seconds for a distance of 100 m. Determine the acceleration of the car.
Above is the graph of the velocity vs. time of a duck flying due south for the winter. At what point might the duck begin reversing directions?
An object can move upward while having a downward acceleration.
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
An object is released from rest near the surface of a planet. The velocity of the object as a function of time is expressed in the following equation. \( v_y = (-3) \, \text{m/s}^2 \, t \) All frictional forces are considered to be negligible. What distance does the object fall \( 10 \) \( \text{s} \) after it is released from rest?
Divers in Acapulco jump from a cliff that is \( 36 \, \text{m} \) above the water with an initial vertical velocity of \( 2 \, \text{m/s} \).
When we refer to an object’s speed, we’re talking about:
A rock is dropped from the top of a tall tower. Half a second later another rock, twice as massive as the first, is dropped. Ignoring air resistance and using ONLY simple kinematics (DO NOT use energy to explain this). Explain it like you would to a 5th grader and select the correct choice:
In a 4.0-kilometer race, a runner completes the first kilometer in 5.9 minutes, the second kilometer in 6.2 minutes, the third kilometer in 6.3 minutes, and the final kilometer in 6.0 minutes. What is the average speed of the runner? Use standard units: m/s.
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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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