| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ m_{\text{bullet}}\, v_i = \left(m_{\text{bullet}} + m_{\text{block}}\right)\, v_x \] | Apply conservation of momentum for this inelastic collision where the bullet embeds in the block. |
| 2 | \[ 0.0350 \times 425 = \left(0.0350 + 0.550\right)\, v_x \] | Substitute the given values: the bullet mass is \(0.0350\) kg, its velocity is \(425\) m/s, and the block mass is \(0.550\) kg. |
| 3 | \[ v_x = \frac{0.0350 \times 425}{0.0350 + 0.550} = \frac{14.875}{0.585} \] | Compute the bullet’s momentum \(0.0350 \times 425 = 14.875\) and the total mass \(0.0350 + 0.550 = 0.585\) kg to solve for \(v_x\). |
| 4 | \[ \boxed{v_x \approx 25.4 \; \text{m/s}} \] | The velocity of the bullet and block together right after the collision is approximately \(25.4\) m/s. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ a = -\mu_k\, g \] | Friction produces a deceleration given by the product of the kinetic friction coefficient \(\mu_k\) and gravitational acceleration \(g\). The negative sign indicates deceleration. |
| 2 | \[ a = -0.40 \times 9.80 = -3.92 \; \text{m/s}^2 \] | Substitute \(\mu_k = 0.40\) and \(g = 9.80 \; \text{m/s}^2\) to calculate the acceleration. |
| 3 | \[ v_x^2 = (25.4)^2 + 2\,(-3.92)\,(10.0) \] | Use the kinematic equation where the initial velocity is the \(25.4\) m/s from part (a) and the displacement \(\Delta x\) is \(10.0\) m. |
| 4 | \[ v_x^2 \approx 645.16 – 78.4 = 566.76 \] | Simplify the expression by computing \((25.4)^2 \approx 645.16\) and \(2 \times 3.92 \times 10.0 = 78.4\). |
| 5 | \[ v_x \approx \sqrt{566.76} \approx 23.8 \; \text{m/s} \] \quad \text{or} \quad \boxed{v_x \approx 23.8 \; \text{m/s}} \] |
Taking the square root yields the final velocity after sliding \(10.0\) m: approximately \(23.8\) m/s. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ m_1\, v_i = \left(m_1 + m_2\right)\, v_x \] | Apply conservation of momentum for the second collision where the bullet-embedded block (\(m_1 = 0.585\) kg) collides inelastically with a stationary block (\(m_2 = 2.50\) kg). |
| 2 | \[ v_x = \frac{0.585 \times 23.8}{0.585 + 2.50} \] | Substitute \(v_i = 23.8\) m/s from part (b) and add the masses \(0.585\) kg and \(2.50\) kg for the collision. |
| 3 | \[ v_x \approx \frac{13.923}{3.085} \approx 4.51 \; \text{m/s} \] | Calculate the post-collision velocity; the numerator \(0.585 \times 23.8 \approx 13.923\) and the total mass is \(3.085\) kg. |
| 4 | \[ 0 = (4.51)^2 + 2\,(-3.92)\,d \] | Use the kinematic equation to find the distance \(d\) traveled before coming to a stop, with \(a = -3.92\) m/s² due to friction. |
| 5 | \[ d = \frac{(4.51)^2}{2 \times 3.92} \] | Solve for \(d\) by rearranging the kinematics equation. |
| 6 | \[ \boxed{d \approx 2.60 \; \text{m}} \] | Evaluating the expression gives a stopping distance of approximately \(2.60\) m after the collision. |
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A pendulum with a period of \( 1 \) \( \text{s} \) on Earth, where the acceleration due to gravity is \( g \), is taken to another planet, where its period is \( 2 \) \( \text{s} \). The acceleration due to gravity on the other planet is most nearly
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A roller coaster ride at an amusement park lifts a car of mass \( 700 \, \text{kg} \) to point \( A \) at a height of \( 90 \, \text{m} \) above the lowest point on the track, as shown above. The car starts from rest at \( A \), rolls with negligible friction down the incline and follows the track around a loop of radius \( 20 \, \text{m} \). Point \( B \), the highest point on the loop, is at a height of \( 50 \, \text{m} \) above the lowest point on the track.
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A block is released from rest and slides down a frictionless ramp inclined at \( 30^\circ \) from the horizontal. When the block reaches the bottom, the block-Earth system has mechanical energy \( \text{E}_i \). The experiment is repeated, but now horizontal and vertical forces of magnitude \( F \) are exerted on the block while it slides, as shown above. When the block reaches the bottom, the mechanical energy of the block-Earth system.
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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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