| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[v_x = \text{constant}\] | The ball leaves the hand with the same horizontal speed as the train because no horizontal forces act on it inside the car. |
| 2 | \[\Delta x = v_x t\] | Horizontal displacement of both ball and thrower grows equally with time. |
| 3 | \[\Delta x_{\text{relative}} = 0\] | Since their \(v_x\) values are identical, their relative horizontal separation is zero; the ball returns to the thrower’s hand. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[v_x(t)=v_i\] | The ball keeps the train’s speed \(v_i\) at release; it experiences no forward force. |
| 2 | \[x_{\text{ball}}=v_i t\] | Ball’s ground‐frame horizontal motion. |
| 3 | \[x_{\text{car}}=v_i t+\tfrac12 a t^{2}\] | Car accelerates with acceleration \(a\) (forward). |
| 4 | \[\Delta x_{\text{relative}}=x_{\text{ball}}-x_{\text{car}}=-\tfrac12 a t^{2}<0\] | Relative displacement is negative (toward rear). Hence the ball lands behind the thrower. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[a_{\text{car}}=-|a|\] | Negative sign denotes slowing down. |
| 2 | \[x_{\text{car}}=v_i t-\tfrac12 |a| t^{2}\] | Car covers less distance than constant‐speed case. |
| 3 | \[\Delta x_{\text{relative}}=v_i t-(v_i t-\tfrac12 |a| t^{2})=\tfrac12 |a| t^{2}>0\] | Positive relative displacement means the ball comes down ahead of the thrower (toward the front). |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[a_{c}=\frac{v^{2}}{R}\] | Car acquires centripetal acceleration toward curve’s center. |
| 2 | \[F_{\text{ball}}=0\;\text{(horizontal)}\] | No horizontal force acts on the ball, so it moves straight (Newton’s 1st law). |
| 3 | \[\text{Ball lands on outer side}\] | The car turns underneath; to riders the ball drifts toward the side opposite the curve’s center — the outer (convex) wall. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[F_{\text{drag}}\propto v_{\text{rel}}^{2}\] | Still outside air produces a backward drag on the rising ball. |
| 2 | \[a_{\text{drag}} < 0\] | Drag decelerates the ball’s horizontal motion relative to the ground. |
| 3 | \[\Delta x_{\text{relative}} < 0\] | Because the train keeps its speed while the ball slows horizontally, it lands behind the thrower. |
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A box with a mass of \( M \) rests on a scale in an elevator that is moving downwards. The elevator slows with an acceleration of \( \dfrac{g}{4} \). Which of the following will give the reading of the scale?
The figure above shows a cart of mass \( M \) accelerating to the right with acceleration \( a \). A block of mass \( m \) is pressed against the cart’s front vertical surface and is held there only by friction. The coefficient of friction between the block and the cart is \( \mu \). What is the minimum acceleration \( a \) of the cart such that the block will not fall?
The steepest street in the world is Baldwin Street in Dunedin, New Zealand. It has an inclination angle of \( 38.0^\circ \) with respect to the horizontal. Suppose a wooden crate with a mass of \( 25.0 \) \( \text{kg} \) is placed on Baldwin Street. An additional force of \( 59 \) \( \text{N} \) must be applied to the crate perpendicular to the pavement in order to hold the crate in place. If the coefficient of static friction between the crate and the pavement is \( 0.599 \), what is the magnitude of the frictional force?
A car travels to right at constant velocity. The net force on the car is
A simple Atwood’s machine remains motionless when equal masses \(M\) are placed on each end of the chord. When a small mass \(m\) is added to one side, the masses have an acceleration \(a\). What is \(M\)? You may neglect friction and the mass of the cord and pulley.
The two blocks of masses \( M \) and \( 2M \) travel at the same speed \( v \) but in opposite directions. They collide and stick together. How much mechanical energy is lost to other forms of energy during the collision?
Four blocks of masses \( 20 \, \text{kg}, \, 30 \, \text{kg}, \, 40 \, \text{kg}, \, \text{and} \, 50 \, \text{kg} \) are stacked on top of one another in an elevator in order of decreasing mass with the lightest mass on the top of the stack. The elevator moves downward with an acceleration of \( 3.2 \, \text{m/s}^2 \). Find the contact force between the \( 30 \, \text{kg} \) and \( 40 \, \text{kg} \) masses.
A skateboarder coasts to a stop on a flat sidewalk. The net force acting on the skateboarder must be ____.
In the diagram above, block \(A\) has a mass of \(3.2 \, \text{kg}\) and block \(B\) a mass of \(2.4 \, \text{kg}\). The pulley is frictionless and has no mass.
Two blocks made of different materials, connected by a thin cord, slide down a plane ramp inclined at an angle \( \theta = 32^\circ \) above the horizontal. If the coefficients of friction are \( \mu_A = 0.2 \) and \( \mu_B = 0.3 \) and if \( m_A = m_B = 5.0 \) \( \text{kg} \), determine:
\(\text{In her hand}\)
\(\text{Behind thrower}\)
\(\text{Ahead of thrower}\)
\(\text{Outer side of curve}\)
\(\text{Behind thrower}\)
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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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