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Step | Derivation/Formula | Reasoning |
---|---|---|
1 | \(\Delta x_1 = v \cdot t\) | Calculate the displacement during the first 1 second. The body moves at a constant speed \(v = 24 \, \text{m/s}\). |
2 | \(\Delta x_1 = 24 \, \text{m/s} \times 1 \, \text{s} = 24 \, \text{m}\) | Substitute \(v = 24 \, \text{m/s}\) and \(t = 1 \, \text{s}\). |
3 | \(v_i = 24 \, \text{m/s}\) | The body continues with initial velocity \(24 \, \text{m/s}\) at \(t = 1 \, \text{s}\), just before deceleration starts. |
4 | \(v_x = v_i + a \cdot \Delta t\) | Calculate the final velocity after 10 seconds of acceleration in the negative \(x\) direction. The acceleration \(a = -6 \, \text{m/s}^2\). |
5 | \(v_x = 24 \, \text{m/s} + (-6 \, \text{m/s}^2 \cdot 10 \, \text{s}) = 24 \, \text{m/s} – 60 \, \text{m/s} = -36 \, \text{m/s}\) | Substitute \(v_i = 24 \, \text{m/s}\), \(a = -6 \, \text{m/s}^2\), and \(\Delta t = 10 \, \text{s}\). |
6 | \(\Delta x_2 = v_i \cdot \Delta t + \frac{1}{2} a \cdot \Delta t^2\) | Calculate the displacement during the 10-second interval of deceleration. |
7 | \(\Delta x_2 = 24 \, \text{m/s} \cdot 10 \, \text{s} + \frac{1}{2} (-6 \, \text{m/s}^2) \cdot (10 \, \text{s})^2\) | Substitute \(v_i = 24 \, \text{m/s}\), \(a = -6 \, \text{m/s}^2\), and \(\Delta t = 10 \, \text{s}\). |
8 | \(\Delta x_2 = 240 \, \text{m} – 300 \, \text{m} = -60 \, \text{m}\) | Simplify the equation to find the displacement during the deceleration period. |
9 | \Delta x = \Delta x_1 + \Delta x_2 | Total displacement is the sum of displacements during different time periods. |
10 | \(\Delta x = 24 \, \text{m} + (-60 \, \text{m}) = -36 \, \text{m}\) | Substitute \(\Delta x_1 = 24 \, \text{m}\) and \(\Delta x_2 = -60 \, \text{m}\). Therefore, the position of the body at \(t = 11 \, \text{seconds}\) is \(\boxed{-36 \, \text{m}}\). |
The correct answer is (c) \(- 36 \, \text{m}\).
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Two balls are dropped off a cliff, 3 seconds apart. The first ball dropped is twice as heavy as the second ball dropped. Air resistance is negligible. While both balls are falling, the distance between the two balls is
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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] | 0.000000000001 |
Nano- | n | [katex]10^{-9}[/katex] | 0.000000001 |
Micro- | µ | [katex]10^{-6}[/katex] | 0.000001 |
Milli- | m | [katex]10^{-3}[/katex] | 0.001 |
Centi- | c | [katex]10^{-2}[/katex] | 0.01 |
Deci- | d | [katex]10^{-1}[/katex] | 0.1 |
(Base unit) | – | [katex]10^{0}[/katex] | 1 |
Deca- or Deka- | da | [katex]10^{1}[/katex] | 10 |
Hecto- | h | [katex]10^{2}[/katex] | 100 |
Kilo- | k | [katex]10^{3}[/katex] | 1,000 |
Mega- | M | [katex]10^{6}[/katex] | 1,000,000 |
Giga- | G | [katex]10^{9}[/katex] | 1,000,000,000 |
Tera- | T | [katex]10^{12}[/katex] | 1,000,000,000,000 |
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