| Step | Reasoning |
|---|---|
| Identify the ratio requested and the expression for energy dissipated. \[ \text{Ratio} = \dfrac{K_i – K_f}{K_i} = 1 – \dfrac{K_f}{K_i} \] |
The question asks for the ratio of mechanical energy dissipated (lost) to the initial kinetic energy. |
| Apply the conservation of momentum to find the final velocity of the combined system. \[ m v_0 = (m + M) v_f \implies v_f = \dfrac{m}{m + M} v_0 \] |
To calculate the final kinetic energy, the speed of the stuck-together blocks must be determined in terms of the initial parameters. |
| Define the initial and final kinetic energies of the system. \[ K_i = \dfrac{1}{2} m v_0^2 \] \[ K_f = \dfrac{1}{2} (m + M) v_f^2 = \dfrac{1}{2} (m + M) \left( \dfrac{m v_0}{m + M} \right)^2 \] |
These expressions are needed to find the ratio of final energy to initial energy. |
| Simplify the expression for the final kinetic energy. \[ K_f = \dfrac{1}{2} (m + M) \dfrac{m^2 v_0^2}{(m + M)^2} = \dfrac{1}{2} \dfrac{m^2 v_0^2}{m + M} \] |
Simplification allows for easier cancellation when calculating the ratio. |
| Calculate the ratio of final kinetic energy to initial kinetic energy. \[ \dfrac{K_f}{K_i} = \dfrac{\dfrac{1}{2} \dfrac{m^2 v_0^2}{m + M}}{\dfrac{1}{2} m v_0^2} = \dfrac{m}{m + M} \] |
This is the fraction of energy remaining, which is the final step before finding the energy lost. |
| Calculate the ratio of energy dissipated. \[ 1 – \dfrac{K_f}{K_i} = 1 – \dfrac{m}{m + M} = \dfrac{m + M – m}{m + M} = \dfrac{M}{m + M} \] |
The energy dissipated is the difference between the initial energy and the remaining energy. |
Why each choice is correct or incorrect:
(A) This choice incorrectly assumes the energy loss ratio is a simple mass ratio without considering the conservation of momentum or the definition of kinetic energy.
(B) This represents the ratio of the kinetic energy remaining in the system after the collision to the initial kinetic energy.
(C) This is the correct answer.
(D) This expression uses a numerator that appears in one-dimensional elastic collision formulas for velocity change, but it is not applicable to the ratio of energy dissipated in an inelastic collision.
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A space probe of mass \(M\) is traveling at a constant velocity \(v_{0}\) through deep space. The probe consists of two modules: a scientific instrument of mass \(0.2M\) and a propulsion unit of mass \(0.8M\). An internal mechanism separates the two modules by pushing them apart with a constant force. After the separation, the scientific instrument is observed to be at rest relative to its original inertial frame of reference. Which of the following correctly describes the velocity of the center of mass of the two-module system after the separation, and provides a correct justification?
A projectile of mass \(m\) is fired horizontally with speed \(v_0\) toward a stationary block of mass \(M\) that is suspended from a rigid support by a light string of length \(L\). The projectile passes through the block and emerges from the other side traveling in the same direction with a speed of \(\alpha v_0\), where \(0 < \alpha < 1\). If the block subsequently swings upward, which of the following expressions represents the maximum height \(h\) reached by the block in terms of the given quantities and fundamental constants?
A block of mass \(m\) moves with speed \(v\) on a horizontal frictionless surface toward a stationary block of mass \(2m\). The stationary block is located at the lowest point of a vertical circular track of radius \(R\). The blocks collide and stick together, then immediately begin to move along the circular track. What is the magnitude of the normal force exerted by the track on the combined blocks immediately after the collision?
Two skaters, Skater X of mass \(M\) and Skater Y of mass \(3M\), are initially at rest on a horizontal, frictionless ice surface. The skaters push off from each other and move in opposite directions. Which of the following correctly compares the final kinetic energy \(K_X\) of Skater X and the final kinetic energy \(K_Y\) of Skater Y after they have separated?
Puck A with mass \(M\) is moving with speed \(v_0\) in the positive \(x\)-direction across a horizontal, frictionless surface. Puck B with mass \(2M\) is moving with speed \(v_0\) in the positive \(y\)-direction. The pucks collide at the origin and stick together, moving as a single object after the collision. What is the ratio of the total kinetic energy of the combined system after the collision to the total kinetic energy of the two-puck system before the collision?
An object of mass \(4M\) is at rest on a horizontal, frictionless surface. An internal explosion splits the object into two fragments of masses \(M\) and \(3M\). Immediately after the explosion, the fragment of mass \(M\) is observed to move to the left with speed \(v\). Which of the following expressions represents the total mechanical energy released during the explosion?
A launcher of total mass M, which includes a projectile of mass m, is initially at rest on a frictionless horizontal table at a height H above the floor. The launcher is positioned at the left edge of the table. The launcher fires the projectile horizontally to the right with a speed v_0 relative to the floor. The launcher recoils to the left, immediately leaves the table, and hits the floor. Which of the following is a correct expression for the horizontal distance x from the left edge of the table to the point where the launcher lands?
Two blocks of mass \(m\) and \(3m\) are held at rest on a horizontal, frictionless surface with a compressed spring of negligible mass between them. The blocks are released and move in opposite directions. Which of the following statements correctly describes the total momentum \(\vec{p}_{sys}\) and total kinetic energy \(K_{sys}\) of the two-block system after the blocks have lost contact with the spring?
A block of mass \(M\) is initially at rest on a horizontal, frictionless surface. A constant horizontal force of magnitude \(F\) is applied to the block for a time interval \(\Delta t\). Immediately after the force is removed, the block slides for an additional time interval \(t\). Which of the following expressions represents the distance \(d\) traveled by the block during the second time interval \(t\)?
Two pucks, each of mass \(M\), slide on a horizontal frictionless surface. Puck 1 moves with speed \(v_1\) in the \(+x\)-direction, and Puck 2 moves with speed \(v_2\) in the \(+y\)-direction. The pucks collide and stick together, moving with a final speed \(v_f\) at an angle \(\theta\) relative to the \(x\)-axis. If \(\tan \theta = \dfrac{3}{4}\), what is the ratio \(\dfrac{v_f}{v_1}\)?
C
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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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