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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | [katex] \Delta x = 100 \, \text{m} [/katex] | The total distance from the car’s position when the light turns red to the stop point is 100 meters. |
| 2 | [katex] v_i = 25 \, \text{m/s} [/katex] | The initial velocity of the car is 25 meters per second. |
| 3 | [katex] v_x = 0 \, \text{m/s} [/katex] | The final velocity of the car is 0 meters per second (the car comes to a halt). |
| 4 | [katex] \mu_k = 0.65 [/katex] | The coefficient of kinetic friction between the car’s tires and the road is 0.65. |
| 5 | [katex] f_k = \mu_k \cdot m \cdot g [/katex] | The kinetic friction force is equal to the coefficient of kinetic friction multiplied by the car’s mass and gravitational acceleration. |
| 6 | [katex] a = -\mu_k \cdot g [/katex] | The acceleration due to friction is the friction force divided by mass (mass cancels out). Here [katex] g [/katex] is the acceleration due to gravity. Using [katex] g = 9.8 \, \text{m/s}^2 [/katex]. |
| 7 | [katex] a = -0.65 \cdot 9.8 \, \text{m/s}^2 = -6.37 \, \text{m/s}^2 [/katex] | Substitute the values into the acceleration formula to find the deceleration. |
| 8 | [katex] v_x^2 = v_i^2 + 2a \Delta x_{braking} [/katex] | Use the kinematic equation to relate the distances and velocities during braking. |
| 9 | [katex] 0 = (25 \, \text{m/s})^2 + 2(-6.37 \, \text{m/s}^2) \Delta x_{braking} [/katex] | Set the final velocity to zero and substitute the initial velocity and acceleration to solve for [katex]\Delta x_{braking}[/katex]. |
| 10 | [katex] 0 = 625 \, \text{m}^2/\text{s}^2 – 12.74 \, \text{m/s}^2 \Delta x_{braking} [/katex] | Simplify the equation. |
| 11 | [katex] \Delta x_{braking} = \frac{625}{12.74} \approx 49.06 \, \text{m} [/katex] | Solve for the braking distance [katex]\Delta x_{braking}[/katex]. |
| 12 | [katex] \Delta x_{reaction} = 100 \, \text{m} – 49.06 \, \text{m} = 50.94 \, \text{m} [/katex] | Subtract the braking distance from the total distance to find the distance covered during reaction time. |
| 13 | [katex] \Delta x_{reaction} = v_i t_{reaction} [/katex] | During the reaction time, the car travels with a constant velocity of 25 m/s. |
| 14 | [katex] 50.94 \, \text{m} = 25 \, \text{m/s} \cdot t_{reaction} [/katex] | Substitute the known values into the reaction time equation. |
| 15 | [katex] t_{reaction} = \frac{50.94 \, \text{m}}{25 \, \text{m/s}} \approx 2.04 \, \text{s} [/katex] | Solve for the reaction time. |
| 16 | [katex] t_{reaction} \approx 2.04 \, \text{s} [/katex] | The reaction time of the driver is approximately [katex]\boxed{2.04 \, \text{s}}[/katex]. |
Just ask: "Help me solve this problem."
A block starts from rest at the top of a 50° incline. The coefficient of kinetic friction between the block and the incline is 0.4. If the block reaches a velocity of 7 m/s at the bottom of the incline, what is the length of the incline?
A large beach ball is dropped from the ceiling of a school gymnasium to the floor about 10 meters below. Which of the following graphs would best represent its velocity as a function of time? (do not neglect air resistance)
A 45 kg crate accelerates at 1.65 m/s2 when pulled by a rope with a force of 200 N. Find the angle the rope is pulled at. Friction is negligible.
A car accelerates from rest with an acceleration of \( 3.5 \, \text{m/s}^2 \) for \( 10 \, \text{s} \). After this, it continues at a constant speed for an unknown amount of time. The driver notices a ramp \( 50 \, \text{m} \) ahead and takes \( 0.6 \, \text{s} \) to react. After reacting, the driver hits the brakes, which slow the car with an acceleration of \( 7.2 \, \text{m/s}^2 \). Unfortunately, the driver does not stop in time and goes off the \( 3 \, \text{m} \) high ramp that is angled at \( 27^\circ \).
2.04 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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