| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[v_{x}^{2}=v_{i}^{2}+2(-g)\Delta x\] | Use the kinematic relation \(v_{x}^{2}=v_{i}^{2}+2a\Delta x\) with upward positive (so \(a=-g\)). \(v_{x}=14\,\text{m/s}\) at the window, \(\Delta x=18\,\text{m}\). |
| 2 | \[v_{i}^{2}=v_{x}^{2}+2g\Delta x\] | Algebraically solve for \(v_{i}^{2}\). |
| 3 | \[v_{i}^{2}=14^{2}+2(9.8)(18)=196+352.8=548.8\] | Substitute the numerical values. |
| 4 | \[\boxed{v_{i}=23.43\,\text{m/s}}\] | Take the square root to obtain the initial speed. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[0=v_{i}^{2}+2(-g)\Delta x_{\text{max}}\] | At the peak, the final velocity is zero, so set \(v_{x}=0\). |
| 2 | \[\Delta x_{\text{max}}=\frac{v_{i}^{2}}{2g}\] | Re-arrange for the upward displacement from the street. |
| 3 | \[\Delta x_{\text{max}}=\frac{548.8}{19.6}=28\,\text{m}\] | Insert \(v_{i}^{2}=548.8\) and \(g=9.8\,\text{m/s}^{2}\). |
| 4 | \[\boxed{28\,\text{m}}\] | The ball rises 28 m above the street. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[v_{x}=v_{i}-g t_{1}\] | Use \(v_{x}=v_{i}+at\) with \(a=-g\) to relate velocities and time. |
| 2 | \[t_{1}=\frac{v_{i}-v_{x}}{g}\] | Solve for \(t_{1}\), the interval from the throw to passing the window upward. |
| 3 | \[t_{1}=\frac{23.5-14}{9.8}=0.964\,\text{s}\] | Insert the numerical values. |
| 4 | \[\boxed{0.96\,\text{s}}\] | The ball was thrown roughly one second before being seen at the window. |
| Step | Derivation / Formula | Reasoning |
|---|---|---|
| 1 | \[T_{\text{total}}=\frac{2v_{i}}{g}\] | For motion that starts and ends at the same height, total flight time is twice the time to the peak, \(v_{i}/g\). |
| 2 | \[T_{\text{total}}=\frac{2(23.5)}{9.8}=4.79\,\text{s}\] | Insert \(v_{i}=23.5\,\text{m/s}\) and \(g=9.8\,\text{m/s}^{2}\). |
| 3 | \[t_{\text{after window}}=T_{\text{total}}-t_{1}=4.79-0.96=3.83\,\text{s}\] | Subtract the elapsed time before passing the window to find the interval after it. |
| 4 | \[\boxed{4.8\,\text{s}\;\text{(total from throw)}}\] | The ball returns to the street 4.8 s after being thrown, i.e., about 3.8 s after passing the window. |
A Major Upgrade To Phy Is Coming Soon — Stay Tuned
We'll help clarify entire units in one hour or less — guaranteed.
A projectile is launched at \( 25 \) \( \text{m/s} \) at an angle of \( 37^{\circ} \). It lands on a platform that is \( 5.0 \) \( \text{m} \) above the launch height.
A person whose weight is \(4.92 \times 10^2 \, \text{N}\) is being pulled up vertically by a rope from the bottom of a cave that is \(35.2 \, \text{m}\) deep. The maximum tension that the rope can withstand without breaking is \(592 \, \text{N}\). What is the shortest time, starting from rest, in which the person can be brought out of the cave?
Two objects are dropped from rest from the same height. Object \( A \) falls through a distance \( d_A \) during a time \( t \), and object \( B \) falls through a distance \( d_B \) during a time \( 2t \). If air resistance is negligible, what is the relationship between \( d_A \) and \( d_B \)?
Which pair of graphs represents the same 1-dimensional motion?
An object is thrown downward at \(23 ~\text{m/s}\) from the top of a \(200 ~\text{m}\) tall building.
A mine shaft is known to be 57.8 m deep. If you dropped a rock down the shaft, how long would it take for you to hear the sound of the rock hitting the bottom of the shaft knowing that sound travels at a constant velocity of 345 m/s?
Which graph below shows that one of the runners started 10 meters further ahead of the other? Assume the y-axis is measured in meters and the x-axis is measured in seconds.
A ball is tossed directly upward. Its total time in the air is \( T \). Its maximum height is \( H \). What is its height after it has been in the air a time \( T/4 \)? Air resistance is negligible.
A projectile of mass 0.750 kg is shot straight up with an initial speed of 18.0 m/s.
\(23.43\,\text{m/s}\)
\(28\,\text{m}\)
\(0.96\,\text{s}\)
\(4.8\,\text{s}\)
By continuing you (1) agree to our Terms of Use and Terms of Sale and (2) consent to sharing your IP and browser information used by this site’s security protocols as outlined in our Privacy Policy.
| 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] |
One price to unlock most advanced version of Phy across all our tools.
per month
Billed Monthly. Cancel Anytime.
We crafted THE Ultimate A.P Physics 1 Program so you can learn faster and score higher.
Try our free calculator to see what you need to get a 5 on the 2026 AP Physics 1 exam.
Understand you mistakes quicker.
Phy automatically provides feedback so you can improve your responses.
10 Free Credits To Get You Started
By continuing you agree to nerd-notes.com Terms of Service, Privacy Policy, and our usage of user data.
Feeling uneasy about your next physics test? We'll boost your grade in 3 lessons or less—guaranteed
