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Velocity just after the collision
Step | Formula Derivation | Reasoning |
---|---|---|
1 | [katex]m_1v_1 + m_2v_2 = (m_1 + m_2)v'[/katex] | Conservation of momentum, where [katex]m_1[/katex] and [katex]m_2[/katex] are the masses of the bullet and block, [katex]v_1[/katex] and [katex]v_2[/katex] are their initial velocities, and [katex]v'[/katex] is their final velocity. |
2 | [katex]v’ = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}[/katex] | Solve for [katex]v'[/katex]. Given: [katex]m_1 = 0.0350 , \text{kg}, v_1 = 425 , \text{m/s}, m_2 = 0.550 , \text{kg}, v_2 = 0 , \text{m/s}[/katex]. |
3 | [katex]v’ = 25.43 , \text{m/s}[/katex] | The velocity just after the collision |
Step | Formula Derivation | Reasoning |
---|---|---|
1 | [katex]v^2 = v’^2 + 2ad[/katex] | Kinematic equation for motion under constant acceleration, where [katex]v[/katex] is the final velocity, [katex]v'[/katex] is the initial velocity, [katex]a[/katex] is acceleration, and [katex]d[/katex] is the distance. |
2 | [katex]a = -\mu_k g[/katex] | Acceleration due to kinetic friction, where [katex]\mu_k[/katex] is the coefficient of kinetic friction and [katex]g[/katex] is the acceleration due to gravity (approximately [katex]9.81 , \text{m/s}^2[/katex]). Given: [katex]\mu_k = 0.40[/katex]. |
3 | [katex]v = \sqrt{v’^2 + 2ad}[/katex] | Solve for [katex]v[/katex]. Given: [katex]v’ = 25.43 , \text{m/s}, d = 10.0 , \text{m}[/katex]. |
4 | [katex]v = 23.84 , \text{m/s}[/katex] | Velocity after sliding 10.0 meters |
Distance traveled by the combined system (2 blocks and the bullet)
Step | Formula Derivation | Reasoning |
---|---|---|
1 | [katex]m_1v_1 + m_2v_2 = (m_1 + m_2)v'[/katex] | Conservation of momentum for the collision between the bullet-block system and the second block, where [katex]m_1[/katex] and [katex]v_1[/katex] are the mass and velocity of the bullet-block system, [katex]m_2[/katex] and [katex]v_2[/katex] are the mass and velocity of the second block, and [katex]v'[/katex] is the final velocity of the combined system. |
2 | [katex]v’ = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}[/katex] | Solve for [katex]v'[/katex]. Given: [katex]m_1 = 0.585 , \text{kg}, v_1 = 23.84 , \text{m/s}, m_2 = 2.50 , \text{kg}, v_2 = 0 , \text{m/s}[/katex]. |
3 | [katex]0 = v’^2 + 2ad[/katex] | Kinematic equation for motion under constant acceleration when the final velocity is 0. |
4 | [katex]d = \frac{-v’^2}{2a}[/katex] | Solve for [katex]d[/katex]. The acceleration [katex]a[/katex] remains [katex]-\mu_k g[/katex] as before. |
5 | [katex]v’ = 4.52 , \text{m/s}[/katex] | Final velocity of the combined system after the second collision |
6 | [katex]d = 2.60 , \text{m}[/katex] | Distance traveled by the combined system before stopping. |
Just ask: "Help me solve this problem."
Consider the following cases of inelastic collisions.
Case (1) – A car moving at 75 mph collides with another car of equal mass moving at 75 mph in the opposite direction and comes to a stop.
Case (2) A car moving at 75 mph hits a stationary steel wall and rolls backs.
The collision time is the same for both cases. In which of these cases would result in the greatest impact force?
Two masses m1 and 4m1 are on an incline. Both surfaces have the same coefficient of kinetic friction. Both objects start from rest, at the same height. Which mass has the largest speed at the bottom?
A car traveling to the right with a speed v brakes to a stop in a distance d. What is the work done on the car by the frictional force F? (Assume that the frictional force is constant)
A 0.5 kg cart, on a frictionless 2 m long table, is being pulled by a 0.1 kg mass connected by a string and hanging over a pulley. The system is released from rest. After the hanging mass falls 0.5 m, calculate the speed of the cart on the table. Use ONLY forces and energy.
A crate is pulled 2.5 m at constant velocity along a 25° incline. The coefficient of kinetic friction between the crate and the plane is 0.250. What is the efficiency of this procedure?
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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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