0 attempts

0% avg

UBQ Credits

Step | Derivation/Formula | Reasoning |
---|---|---|

(a) How high does the ball rise? |
||

1 | Use the kinematic equation: [katex]v_f^2 = v_i^2 + 2a d[/katex] | This equation relates the final velocity, initial velocity, acceleration, and distance traveled. |

2 | [katex]0 = (22.5 \, \text{m/s})^2 + 2(-9.8 \, \text{m/s}^2)d[/katex] | At the highest point, the final velocity [katex]v_f = 0[/katex] m/s. The initial velocity [katex]v_i = 22.5[/katex] m/s and acceleration [katex]a = -9.8[/katex] m/s[katex]^2[/katex] (gravity acting downwards). |

3 | [katex]d = \frac{(22.5 \, \text{m/s})^2}{2 \cdot 9.8 \, \text{m/s}^2} \approx 25.8 \, \text{m}[/katex] | Solve for [katex]d[/katex], the height reached by the ball. |

4 | [katex]d \approx 25.8 \, \text{m}[/katex] |
The maximum height the ball reaches. |

(b) How long does it take for the ball to reach its highest point? |
||

1 | Use the kinematic equation: [katex]v_f = v_i + at[/katex] | This equation relates the final velocity, initial velocity, acceleration, and time. |

2 | 0 = 22.5 \, \text{m/s} + (-9.8 \, \text{m/s}^2) t | At the highest point, [katex]v_f = 0[/katex] m/s. The initial velocity [katex]v_i = 22.5[/katex] m/s and acceleration [katex]a = -9.8[/katex] m/s[katex]^2[/katex]. |

3 | t = \frac{22.5 \, \text{m/s}}{9.8 \, \text{m/s}^2} \approx 2.30 \, \text{s} | Solve for [katex]t[/katex], the time taken to reach the highest point. |

4 | t \approx 2.30 \, \text{s} |
The time taken to reach the maximum height. |

(c) How long does the ball remain in the air? |
||

1 | \text{Total time} = 2 \times t_{\text{up}} | Time to go up is equal to time to come down because distances and accelerations are the same. |

2 | 2 \times 2.30 \, \text{s} \approx 4.60 \, \text{s} | Double the time to reach the highest point to find total time in the air. |

3 | \text{Total time} \approx 4.60 \, \text{s} |
The total time the ball remains in the air. |

(d) How fast was it going just before it is caught? |
||

1 | Use symmetry: [katex]v_{\text{final}} = -v_{\text{initial}}[/katex] | Due to symmetry, speed upon returning to original height equals initial speed but opposite in direction. |

2 | \text{Speed} = 22.5 \, \text{m/s} | The magnitude of the speed is the same. |

3 | \text{Speed} = 22.5 \, \text{m/s} |
The speed just before being caught. |

(e) What is the velocity and acceleration of the ball at the highest point? |
||

1 | \text{Velocity} = 0 \, \text{m/s} | At the highest point, the velocity is zero as the ball changes direction. |

2 | \text{Acceleration} = -9.8 \, \text{m/s}^2 | Acceleration due to gravity remains constant throughout the motion. |

3 | \text{Velocity} = 0 \, \text{m/s}, \text{Acceleration} = -9.8 \, \text{m/s}^2 |
Velocity and acceleration at the highest point. |

Just ask: "Help me solve this problem."

- Statistics

Advanced

Mathematical

GQ

A car is driving at \(25 \, \text{m/s}\) when a light turns red \(100 \, \text{m}\) ahead. The driver takes an unknown amount of time to react and hit the brakes, but manages to skid to a stop at the red light. If \(\mu_s = 0.9\) and \(\mu_k = 0.65\), what was the reaction time of the driver?

- 1D Kinematics, Friction, Linear Forces

Intermediate

Mathematical

FRQ

You throw a rock straight up with an initial velocity of \( 5.0 \, \text{m/s} \).

- 1D Kinematics

Advanced

Mathematical

MCQ

Ball 1 is dropped from rest at time \( t = 0 \) from a tower of height \( h \). At the same instant, ball 2 is launched upward from the ground with the initial speed \( v_0 \). If air resistance is negligible, at what time \( t \) will the two balls pass each other?

- 1D Kinematics, Free Fall

Intermediate

Conceptual

MCQ

Which of the following graphs represent an object having zero acceleration? (There could be more than one answer)

- Motion Graphs

Advanced

Mathematical

MCQ

An object undergoes constant acceleration. Starting from rest, the object travels \( 5 \, \text{m} \) in the first second. Then it travels \( 15 \, \text{m} \) in the next second. What additional distance will be covered in the third second?

- 1D Kinematics

- \( 25.8 \, \text{m} \)
- \( 2.3 \, \text{s} \)
- \( 4.6 \, \text{s} \)
- \( 22.5 \, \text{m/s} \)
- \( 0 \, \text{m/s} , \, -9.8 \, \text{m/s}^2 \)

By continuing you (1) agree to our Terms of Sale and Terms of Use 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] |

General Metric Conversion Chart

Conversion Example

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 |

- 1. Some answers may vary by 1% due to rounding.
- Gravity values may differ: \(9.81 \, \text{m/s}^2\) or \(10 \, \text{m/s}^2\).
- Variables can be written differently. For example, initial velocity (\(v_i\)) may be \(u\), and displacement (\(\Delta x\)) may be \(s\).
- Bookmark questions you can’t solve to revisit them later
- 5. Seek help if you’re stuck. The sooner you understand, the better your chances on tests.

The most advanced version of Phy. 50% off, for early supporters. Prices increase soon.

per month

Billed Monthly. Cancel Anytime.

Trial –> Phy Pro

- Unlimited Messages and Images
- Unlimited UBQ Credits
- 157% Better than GPT
- 30 --> 300 Word Input
- 3 --> 15 MB Image Size Limit
- 1 --> 3 Images per Message
- All Smart Actions
- Mobile Snaps
- Focus Mode
- No Ads

A quick explanation

Credits are used to grade your FRQs and GQs. Pro users get unlimited credits.

Submitting counts as 1 attempt.

Viewing answers or explanations count as a failed attempts.

Phy gives partial credit if needed

MCQs and GQs are are 1 point each. FRQs will state points for each part.

Phy customizes problem explanations based on what you struggle with. Just hit the explanation button to see.

Understand you mistakes quicker.

Phy automatically provides feedback so you can improve your responses.

10 Free Credits To Get You Started