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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \(v_x = v_i + at_1\) | First, use the equation of motion to express final velocity \(v_x\) at \(t_1\). Since the sled starts from rest, \(v_i = 0\), so \(v_x = at_1\). |
| 2 | \(v_x = 13.0 \, \text{m/s}^2 \times t_1\) | Substitute the given acceleration \(a = 13.0 \, \text{m/s}^2\). |
| 3 | \(d_1 = \frac{1}{2} a t_1^2\) | Use the equation for displacement under constant acceleration to find \(d_1\), the distance covered during the acceleration phase. |
| 4 | \(d_1 = \frac{1}{2} \times 13.0 \, \text{m/s}^2 \times t_1^2\) | Substitute the value of \(a\). |
| 5 | \(d_2 = v_x(t_2 – t_1)\) | Use the equation for displacement under constant velocity for the distance traveled from \(t_1\) to \(t_2\). |
| 6 | \(d_2 = \left(13.0 \, \text{m/s}^2 \times t_1\right)\left(90.0\, \text{s} – t_1\right)\) | Substitute \(v_x\) and the time intervals into the equation for the second part of the journey. |
| 7 | \(d_1 + d_2 = 5300 \, \text{m}\) | Use the total distance formula, combining both displacements to equal the total given distance. |
| 8 | \(\frac{1}{2} \times 13.0 \, \text{m/s}^2 \times t_1^2 + (13.0 \, \text{m/s}^2 \times t_1)(90.0 \, \text{s} – t_1) = 5300 \, \text{m}\) | Combine the equations for \(d_1\) and \(d_2\). |
| 9 | \(6.5 t_1^2 + 1170 t_1 – 13 t_1^2 = 5300\) | Simplify the equation by combining like terms. |
| 10 | \(-6.5 t_1^2 + 1170 t_1 – 5300 = 0\) | Combine like terms and solve the quadratic equation via the quadratic formula or graphing. |
| 11 | \(t_1 = 4.65 \, \text{s}\) | Final answer. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \(t_2 + t_1 = 90.0 \, \text{s}\) | The total time of travel is the time of the \(t_1\) and \(t_2\) added together. |
| 2 | \(t_2 = 90.0 \, \text{s} – t_1 \, \text{s}\) | Rearrange and solve for \(t_2\). |
| 3 | \(t_2 = 90.0 \, \text{s} – 4.58 \, \text{s}\) | Substitute the calculated value of \(t_1 \) found in part a. |
| 4 | \(t_2 = 85.35 \, \text{s}\) | Final \(t_2\) value. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \(v = a t_1\) | Use the equation for final velocity under constant acceleration. This comes from the kinematic formula \(v = v_0 + a t_1\), where \(v_0\) is zero. |
| 2 | \(v = 13.0 \, \text{m/s}^2 \times 4.65 \, \text{s}\) | Substitute the value of \(a\) and \(t_1\) into the equation. |
| 3 | \(v = 60.5 \, \text{m/s}\) | Calculate the velocity. |
– (a) \( \boxed{t_1 = 4.58 \, \text{s}} \)
– (b) \( \boxed{t_2 = 85.42\, \text{s}} \)
– (c) \( \boxed{v = 59.54 \, \text{m/s}} \)
Just ask: "Help me solve this problem."
At time \( t = 0 \), a cart is at \( x = 10 \, \text{m} \) and has a velocity of \( 3 \, \text{m/s} \) in the \( -x \) direction. The cart has a constant acceleration in the \( +x \) direction with magnitude \( 3 \, \text{m/s}^2 < a < 6 \, \text{m/s}^2 \). Which of the following gives the possible range of the position of the cart at \( t = 1 \, \text{s} \)?
Two identical metal balls are being held side by side at the top of a ramp. Alex lets one ball, \( A \), start rolling down the hill. A few seconds later, Alex’s partner, Bob, starts the second ball, \( B \), down the hill by giving it a push. Ball \( B \) rolls down the hill along a line parallel to the path of the first ball and passes it. At the instant ball \( B \) passes ball \( A \):
Two students are on a balcony 19.6 m above the street. One student throws a ball vertically downward at 14.7 m/s. At the same instant, the other student throws a ball vertically upward at the same speed. The second ball just misses the balcony on the way down.
A projectile of mass 0.750 kg is shot straight up with an initial speed of 18.0 m/s.
A 25 g steel ball is attached to the top of a 24-cm-diameter vertical wheel. Starting from rest, the wheel accelerates at [katex] 470 \, \frac{rad}{s^2}[/katex]. The ball is released after [katex]\frac{3}{4} [/katex] of a revolution. How high does it go above the center of the wheel?
\( t_1 = 4.65 \, \text{s} \)
\( t_2 = 85.35 \, \text{s} \)
\( v = 60.5 \, \text{m/s} \)
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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] |
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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