Two students are on a balcony 19.6 m above the street. One student throws a ball vertically downward at 14.7 m/s. At the same instant, the other student throws a ball vertically upward at the same speed. The second ball just misses the balcony on the way down.

- (a) What is the difference in the time the balls spend in the air?
*(2 points)* - (b) What is the velocity of each ball as it strikes the ground?
*(2 points)* - (c) How far apart are the balls 0.800 s after they are thrown?
*(3 points)*

**Difference in Time Spent in the Air**: \boxed{\Delta t = 0.5 , \text{seconds}}**Velocity of Each Ball as it Strikes the Ground**:- Ball Thrown Downward: \boxed{v_{\text{down}} = 53.9 , \text{m/s}}
- Ball Thrown Upward: \boxed{v_{\text{up}} = 19.6 , \text{m/s}}

**Distance Apart after 0.800 Seconds**: \boxed{\text{Distance Apart} = 2.352 , \text{meters}}

**Difference in Time Spent in the Air**:

Step Formula Derivation Reasoning 1 H = v_0t_{\text{down}} + \frac{1}{2}gt_{\text{down}}^2 Kinematic equation for downward motion; solve for t_{\text{down} 2 t_{\text{up-to-balcony}} = \frac{v_0}{g} Time to reach the highest point and back to the balcony height for upward motion 3 H = \frac{1}{2}gt_{\text{up-additional}}^2 Additional time from balcony to ground for upward motion 4 \Delta t = t_{\text{down}} – t_{\text{up}} Difference in time spent in the air 5 \boxed{\Delta t = 0.5 , \text{seconds}} Final Calculation **Velocity of Each Ball as it Strikes the Ground**:

Ball Thrown Downward Ball Thrown Upward Reasoning v_{\text{down}} = v_0 + gt_{\text{down}} v_{\text{up}} = \sqrt{2gH} Velocity calculation at impact for both balls \boxed{v_{\text{down}} = 53.9 , \text{m/s}} \boxed{v_{\text{up}} = 19.6 , \text{m/s}} Final combination **Distance Apart after 0.800 Seconds**:

Ball Thrown Downward Ball Thrown Upward Reasoning y_{\text{down}} = v_0t – \frac{1}{2}gt^2 y_{\text{up}} = H – (v_0t – \frac{1}{2}gt^2) Position calculation after 0.800 s; distance apart

Phy can also check your working. Just snap a picture!

- Statistics

Beginner

Mathematical

GQ

A car travels at 20 m/s for 5 minutes and then travels another 2 km at 40 m/s. What is the total

distance traveled and time of travel for the car?

- 1D Kinematics

Beginner

Conceptual

MCQ

The displacement x of an object moving in one dimension is shown above as a function of time t. The velocity of this object must be

- 1D Kinematics, Motion Graphs

Advanced

Mathematical

GQ

A car is driving at 25 m/s when a light turns red 100 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 μ_{s}=0.9 and μ_{k}=0.65, what was the reaction time of the driver?

- 1D Kinematics

Intermediate

Conceptual

MCQ

Two balls are dropped off a cliff, 3 seconds apart. The first ball dropped is twice as heavy as the second ball dropped. Air resistance is negligible. While both balls are falling, the distance between the two balls is

- 1D Kinematics, Free Fall

Intermediate

Mathematical

GQ

On a strange, airless planet, a ball is thrown downward from a height of 17 m. The ball initially travels at 15 m/s. If the ball hits the ground in 1 s, what is this planet’s gravitational acceleration?

- 1D Kinematics

Intermediate

Conceptual

GQ

A 100-pound rock and a 1-pound metal arrow pointed downwards, are dropped from height h. Assuming there is no air resistance, which one hits the ground first and why?

- 1D Kinematics, Free Fall

Intermediate

Mathematical

FRQ

A projectile of mass 0.750 kg is shot straight up with an initial speed of 18.0 m/s.

- 1D Kinematics, Energy, Free Fall

Intermediate

Mathematical

GQ

A horizontal spring with spring constant 162 N/m is compressed 50 cm and used to launch a 3 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the rough surface is 0.2. Find the total distance the box travels before stopping.

- 1D Kinematics, Energy, Linear Forces

Beginner

Mathematical

GQ

A car is traveling 20 m/s when the driver sees a child standing on the road. She takes 0.8 s to react then steps on the brakes and slows at 7.0 m/s^{2}. How far does the car go before it stops?

- 1D Kinematics

Advanced

Mathematical

FRQ

A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s.

- 1D Kinematics

**Difference in Time Spent in the Air**: \boxed{\Delta t = 0.5 , \text{seconds}}**Velocity of Each Ball as it Strikes the Ground**:- Ball Thrown Downward: \boxed{v_{\text{down}} = 53.9 , \text{m/s}}
- Ball Thrown Upward: \boxed{v_{\text{up}} = 19.6 , \text{m/s}}

**Distance Apart after 0.800 Seconds**: \boxed{\text{Distance Apart} = 2.352 , \text{meters}}

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_1m_2}{r^2} |

a = \frac{\Delta v}{\Delta t} | f = \mu N |

R = \frac{v_i^2 \sin(2\theta)}{g} |

Circular Motion | Energy |
---|---|

F_c = \frac{mv^2}{r} | KE = \frac{1}{2} mv^2 |

a_c = \frac{v^2}{r} | PE = mgh |

KE_i + PE_i = KE_f + PE_f |

Momentum | Torque and Rotations |
---|---|

p = m v | \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 |
---|

F = -k x |

T = 2\pi \sqrt{\frac{l}{g}} |

T = 2\pi \sqrt{\frac{m}{k}} |

Constant | Description |
---|---|

g | Acceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface |

G | Universal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2 |

\mu_k and \mu_s | Coefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion. |

k | Spring constant, in \text{N/m} |

M_E = 5.972 \times 10^{24} , \text{kg} | Mass of the Earth |

M_M = 7.348 \times 10^{22} , \text{kg} | Mass of the Moon |

M_M = 1.989 \times 10^{30} , \text{kg} | Mass of the Sun |

Variable | SI Unit |
---|---|

s (Displacement) | \text{meters (m)} |

v (Velocity) | \text{meters per second (m/s)} |

a (Acceleration) | \text{meters per second squared (m/s}^2\text{)} |

t (Time) | \text{seconds (s)} |

m (Mass) | \text{kilograms (kg)} |

Variable | Derived SI Unit |
---|---|

F (Force) | \text{newtons (N)} |

E, PE, KE (Energy, Potential Energy, Kinetic Energy) | \text{joules (J)} |

P (Power) | \text{watts (W)} |

p (Momentum) | \text{kilogram meters per second (kgm/s)} |

\omega (Angular Velocity) | \text{radians per second (rad/s)} |

\tau (Torque) | \text{newton meters (Nm)} |

I (Moment of Inertia) | \text{kilogram meter squared (kgm}^2\text{)} |

f (Frequency) | \text{hertz (Hz)} |

General Metric Conversion Chart

Conversion Example

Example of using unit analysis: Convert 5 kilometers to millimeters.

Start with the given measurement:

`\text{5 km}`

Use the conversion factors for kilometers to meters and meters to millimeters:

`\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}`

Perform the multiplication:

`\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}`

Simplify to get the final answer:

`\boxed{5 \times 10^6 \, \text{mm}}`

Prefix | Symbol | Power of Ten | Equivalent |
---|---|---|---|

Pico- | p | 10^{-12} | 0.000000000001 |

Nano- | n | 10^{-9} | 0.000000001 |

Micro- | µ | 10^{-6} | 0.000001 |

Milli- | m | 10^{-3} | 0.001 |

Centi- | c | 10^{-2} | 0.01 |

Deci- | d | 10^{-1} | 0.1 |

(Base unit) | – | 10^{0} | 1 |

Deca- or Deka- | da | 10^{1} | 10 |

Hecto- | h | 10^{2} | 100 |

Kilo- | k | 10^{3} | 1,000 |

Mega- | M | 10^{6} | 1,000,000 |

Giga- | G | 10^{9} | 1,000,000,000 |

Tera- | T | 10^{12} | 1,000,000,000,000 |

- Some answers may be slightly off by 1% depending on rounding, etc.
- Answers will use different values of gravity. Some answers use 9.81 m/s
^{2}, and other 10 m/s^{2 }for calculations. - Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
- Bookmark questions that you can’t solve so you can come back to them later.
- Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

The most advanced version of Phy. Currently 50% off, for early supporters.

per month

Billed Monthly. Cancel Anytime.

Trial –> Phy Pro

- Unlimited Messages
- Unlimited Image Uploads
- Unlimited Smart Actions
- 30 --> 300 Word Input
- 3 --> 15 MB Image Size Limit
- 1 --> 3 Images per Message
- 200% Memory Boost
- 150% Better than GPT
- 75% More Accurate, 50% Faster
- Mobile Snaps
- Focus Mode
- No Ads