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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.
| 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 |
| 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 |
| 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 |
The graph above shows velocity v versus time t for an object in linear motion. Which of the following is a possible graph of position x versus time t for this object?
Below is the graph of the velocity vs. time of a duck flying due south for the winter. At what point might the duck begin reversing directions?
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)
Traveling at a speed of 15.9 m/s, the driver of an automobile suddenly locks the wheels by slamming on the brakes. The coefficient of kinetic friction between the tires and the road is 0.659. What is the speed of the automobile after 1.59 s have elapsed? Ignore the effects of air resistance.
The alarm at a fire station rings and a 79.34-kg fireman, starting from rest, slides down a pole to the floor below (a distance of 4.20 m). Just before landing, his speed is 1.36 m/s. What is the magnitude of the kinetic frictional force exerted on the fireman as he slides down the pole?
A person whose weight is 4.92 × 102 N is being pulled up vertically by a rope from the bottom of a cave that is 35.2 m deep. The maximum tension that the rope can withstand without breaking is 592 N. What is the shortest time, starting from rest, in which the person can be brought out of the cave?
Above is a graph of the distance vs. time for car moving along a road. According the graph, at which of the following times would the automobile have been accelerating positively?
A baseball is thrown vertically into the air with a velocity v, and reaches a maximum height h. At what height was the baseball moving with one-half its original velocity? Assume air resistance is negligible.
A 135.0 N force is applied to a 30.0 kg box at 42 degree angle to the horizontal. If the force of friction is 85.0, what is the net force and acceleration? If the object starts from rest, how far has it traveled in 3.3 sec?
You stand at the edge of a vertical cliff and throws a stone vertically upwards. The stone leaves your hand with a speed v = 8.0 m/s. The time between the stone leaving your hand and hitting the sea is 3.0 s. Assume air resistance is negligible. Calculate:
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?
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.
A rocket is sent to shoot down an invading spacecraft that is hovering at an altitude of 1500 meters. The rocket is launched with an initial velocity of 180 m/s. Find the following:
A 1100 kg car accelerates from 32 m/s to 8.0 m/s in 4.0 sec. What amount of force was needed to slow it down?
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 whiffle ball is tossed straight up, reaches a highest point, and falls back down. Air resistance is not negligible. Which of the following statements are true?
Can an object’s average velocity equal zero when object’s speed is greater than zero?
A rocket, initially at rest, is fired vertically upward with an acceleration of 12.0 m/s2. At an altitude of 1.00 km, the rocket engine cuts off. Drag is negligible.
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/s2. How far does the car go before it stops?
At time t = 0, a cart is at x = 10 m and has a velocity of 3 m/s in the –x direction. The cart has a constant acceleration in the +x direction with magnitude 3 m/s2 < a < 6 m/s2?. Which of the following gives the possible range of the position of the cart at t = 1 s?
Two identical metal balls are being held side by side at the top of a ramp. Alex lets one ball, 4, start rolling down the hill. A few seconds later, Alex’ partner, Bob starts the second ball, B, down the hill by giving it a push. Ball B rolls down the hill along a line parallel to the path of the first ball and passes it. At the instant ball B passes ball A:
In a 4.0-kilometer race, a runner completes the first kilometer in 5.9 minutes, the second kilometer in 6.2 minutes, the third kilometer in 6.3 minutes, and the final kilometer in 6.0 minutes. What is the average speed of the runner? Use standard units: m/s.
An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at +13.0 m/s2. At t1 the rocket engine is shut down and the sled moves with constant velocity v until t2. The total distance traveled by the sled is 5.30 x 103 m and the total time is 90.0 s. Find t1, t2, and v.
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?
A ranger in a national park is driving at 56 km/h when a deer jumps onto the road 65 m ahead of the vehicle. After a reaction time of t s, the ranger applies the brakes to produce an acceleration of -3 m/s2. What is the maximum reaction time allowed if the ranger is to avoid hitting the deer?
A coin is dropped from a hot air-balloon that is 250 m above the ground rising at 11 m/s upwards. For the coin, find the following:
Priscilla the Penguin stands at the edge of a rock ledge and tosses a small ice cube directly upward with an initial velocity of vo. The ice cube’s initial height above the ground is 3.25 m, and reaches it maximum height above the ground 0.586 s after being thrown. The ice cube then plummets to the ground, missing the edge of the rock ledge on its way down.
Which of the following statements about the acceleration due to gravity is TRUE?
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| Kinematics | Forces |
|---|---|
| \Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2 | F = m \cdot a |
| v = v_i + a \cdot t | F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} |
| a = \frac{\Delta v}{\Delta t} | f = \mu \cdot N |
| R = \frac{v_i^2 \cdot \sin(2\theta)}{g} |
| Circular Motion | Energy |
|---|---|
| F_c = \frac{m \cdot v^2}{r} | KE = \frac{1}{2} m \cdot v^2 |
| a_c = \frac{v^2}{r} | PE = m \cdot g \cdot h |
| KE_i + PE_i = KE_f + PE_f |
| Momentum | Torque and Rotations |
|---|---|
| p = m \cdot v | \tau = r \cdot F \cdot \sin(\theta) |
| J = \Delta p | I = \sum m \cdot r^2 |
| p_i = p_f | L = I \cdot \omega |
| Simple Harmonic Motion |
|---|
| F = -k \cdot 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} |
| 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 (kg·m/s)} |
| \omega (Angular Velocity) | \text{radians per second (rad/s)} |
| \tau (Torque) | \text{newton meters (N·m)} |
| I (Moment of Inertia) | \text{kilogram meter squared (kg·m}^2\text{)} |
| f (Frequency) | \text{hertz (Hz)} |
General Metric Conversion Chart
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} | |
Nano- | n | 10^{-9} | |
Micro- | µ | 10^{-6} | |
Milli- | m | 10^{-3} | |
Centi- | c | 10^{-2} | |
Deci- | d | 10^{-1} | |
(Base unit) | – | 10^{0} | |
Deca- or Deka- | da | 10^{1} | |
Hecto- | h | 10^{2} | |
Kilo- | k | 10^{3} | |
Mega- | M | 10^{6} | |
Giga- | G | 10^{9} | |
Tera- | T | 10^{12} |
