Question: Prove these statements: (a) As long as

> A baseball thrown at an angle of 60.0° above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the ball’s initial velocity (the velocity with which th

> An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at 75 m/s in a direction 55° above the horizontal. How far in front of the ca

> According to Guinness World Records, the longest home run ever measured was hit by Roy “Dizzy” Carlyle in a minor league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) If the ball’s initial velocity was in a

> An important piece of landing equipment must be thrown to a ship, which is moving at 45.0 cm>s, before the ship can dock. This equipment is thrown at 15.0 m>s at 60.0° above the horizontal from the top of a tower at the edge of the water

> A toy rocket is launched with an initial velocity of 12.0 m>s in the horizontal direction from the roof of a 30.0-m-tall building. The rocket’s engine produces a horizontal acceleration of (1.60 m/s3)t, in the same direction as the initial velocity, but

> You are on the west bank of a river that is flowing north with a speed of 1.2 m/s. Your swimming speed relative to the water is 1.5 m/s, and the river is 60 m wide. What is your path relative to the earth that allows you to cross the river in the shortes

> A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at 43.0° above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the t

> An airplane is flying with a velocity of 90.0 m/s at an angle of 23.0° above the horizontal. When the plane is 114 m directly above a dog that is standing on level ground, a suitcase drops out of the luggage compartment. How far from the dog will the sui

> A model car starts from rest and travels in a straight line. A smartphone mounted on the car has an app that transmits the magnitude of the car’s acceleration (measured by an accelerometer) every second. The results are given in the tab

> In a physics lab experiment, you release a small steel ball at various heights above the ground and measure the ball’s speed just before it strikes the ground. You plot your data on a graph that has the release height (in meters) on the vertical axis and

> In your physics lab you release a small glider from rest at various points on a long, frictionless air track that is inclined at an angle u above the horizontal. With an electronic photocell, you measure the time t it takes the glider to slide a distance

> Cars A and B travel in a straight line. The distance of A from the starting point is given as a function of time by xA(t)= at + bt2, with a = 2.60 m/s and b = 1.20 m/s2. The distance of B from the starting point is xB(t)= gt2 - dt3, with g = 2.80 m/s2 an

> A ball is thrown straight up from the ground with speed v0. At the same instant, a second ball is dropped from rest from a height H, directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which t

> An object is moving along the x-axis. At t = 0 it has velocity v0x = 20.0 m/s. Starting at time t = 0 it has acceleration ax = -Ct, where C has units of m/s3. (a) What is the value of C if the object stops in 8.00 s after t = 0? (b) For the value of C ca

> You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.00 s later you hear the sound of the rock hitting the ground at the foot of the cli

> A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 m/s2. Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of

> Raindrops hitting the side windows of a car in motion often leave diagonal streaks even if there is no wind. Why? Is the explanation the same or different for diagonal streaks on the windshield?

> A physics teacher performing an outdoor demonstration suddenly falls from rest off a high cliff and simultaneously shouts “Help.” When she has fallen for 3.0 s, she hears the echo of her shout from the valley floor below. The speed of sound is 340 m/s. (

> During your summer internship for an aerospace company, you are asked to design a small research rocket. The rocket is to be launched from rest from the earth’s surface and is to reach a maximum height of 960 m above the earth’s surface. The rocket’s eng

> In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of 3.50 m/s2 upward. At 25.0 s after launch, the second stage fires for 10.0 s, which boosts the rocket’s velocity to 13

> Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a

> A flowerpot falls off a windowsill and passes the window of the story below. Ignore air resistance. It takes the pot 0.380 s to pass from the top to the bottom of this window, which is 1.90 m high. How far is the top of the window below the windowsill fr

> Sam heaves a 16-lb shot straight up, giving it a constant upward acceleration from rest of 35.0 m/s2 for 64.0 cm. He releases it 2.20 m above the ground. Ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above th

> An entertainer juggles balls while doing other activities. In one act, she throws a ball vertically upward, and while it is in the air, she runs to and from a table 5.50 m away at an average speed of 3.00 m/s, returning just in time to catch the falling

> A certain volcano on earth can eject rocks vertically to a maximum height H. (a) How high (in terms of H) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is 3.71 m/s2; ignore

> You are on the roof of the physics building, 46.0 m above the ground (Fig. P2.70). Your physics professor, who is 1.80 m tall, is walking alongside the building at a constant speed of 1.20 m/s. If you wish to drop an egg on your professorâ€&#1

> The acceleration of a particle is given by ax(t) = -2.00 m/s2 +(3.00 m/s3)t. (a) Find the initial velocity v0x such that the particle will have the same x-coordinate at t = 4.00 s as it had at t = 0. (b) What will be the velocity at t = 4.00 s ?

> In uniform circular motion, what are the average velocity and average acceleration for one revolution? Explain.

> An object’s velocity is measured to be vx(t) = a - bt2, where a = 4.00 m/s and b = 2.00 m/s3. At t = 0 the object is at x = 0. (a) Calculate the object’s position and acceleration as functions of time. (b) What is the object’s maximum positive displaceme

> The driver of a car wishes to pass a truck that is traveling at a constant speed of 20.0 m/s (about 45 mi/h). Initially, the car is also traveling at 20.0 m/s, and its front bumper is 24.0 m behind the truck’s rear bumper. The car accelerates at a consta

> You are standing at rest at a bus stop. A bus moving at a constant speed of 5.00 m/s passes you. When the rear of the bus is 12.0 m past you, you realize that it is your bus, so you start to run toward it with a constant acceleration of 0.960 m/s2. How f

> A car and a truck start from rest at the same instant, with the car initially at some distance behind the truck. The truck has a constant acceleration of 2.10 m/s2, and the car has an acceleration of 3.40 m/s2. The car overtakes the truck after the truck

> Two cars start 200 m apart and drive toward each other at a steady 10 m/s. On the front of one of them, an energetic grasshopper jumps back and forth between the cars (he has strong legs!) with a constant horizontal velocity of 15 m/s relative to the gro

> A ball starts from rest and rolls down a hill with uniform acceleration, traveling 200 m during the second 5.0 s of its motion. How far did it roll during the first 5.0 s of motion?

> The engineer of a passenger train traveling at 25.0 m/s sights a freight train whose caboose is 200 m ahead on the same track (Fig. P2.62). The freight train is traveling at 15.0 m/s in the same direction as the passenger train. The engineer of the passe

> A gazelle is running in a straight line (the x-axis). The graph in Fig. P2.61 shows this animal’s velocity as a function of time. During the first 12.0 s, find (a) the total distance moved and (b) the displacement of the gazelle. (c) Sk

> A subway train starts from rest at a station and accelerates at a rate of 1.60 m/s2 for 14.0 s. It runs at constant speed for 70.0 s and slows down at a rate of 3.50 m/s2 until it stops at the next station. Find the total distance covered.

> A rocket carrying a satellite is accelerating straight up from the earth’s surface. At 1.15 s after liftoff, the rocket clears the top of its launch platform, 63 m above the ground. After an additional 4.75 s, it is 1.00 km above the ground. Calculate th

> When you drop an object from a certain height, it takes time T to reach the ground with no air resistance. If you dropped it from three times that height, how long (in terms of T) would it take to reach the ground?

> A brick is dropped from the roof of a tall building. After it has been falling for a few seconds, it falls 40.0 m in a 1.00-s time interval. What distance will it fall during the next 1.00 s? Ignore air resistance.

> Earthquakes produce several types of shock waves. The most well-known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth’s crust, P-waves travel at about 6.5 km/s and S-waves move at about 3.5 km/s. The t

> A lunar lander is descending toward the moon’s surface. Until the lander reaches the surface, its height above the surface of the moon is given by y(t)= b - ct + dt2, where b = 800 m is the initial height of the lander above the surface, c = 60.0 m/s, an

> A typical male sprinter can maintain his maximum acceleration for 2.0 s, and his maximum speed is 10 m/ s. After he reaches this maximum speed, his acceleration becomes zero, and then he runs at constant speed. Assume that his acceleration is constant du

> The Mars Polar Lander spacecraft was launched on January 3, 1999. On December 3, 1999, the day Mars Polar Lander impacted the Martian surface at high velocity and probably disintegrated, the positions of the earth and Mars were given by these coordinates

> You are a mechanical engineer working for a manufacturing company. Two forces, F S 1 and F S 2, act on a component part of a piece of equipment. Your boss asked you to find the magnitude of the larger of these two forces. You can vary the angle between F

> You are a team leader at a pharmaceutical company. Several technicians are preparing samples, and you want to compare the densities of the samples (density = mass>volume) by using the mass and volume values they have reported. Unfortunately, you did n

> Later in our study of physics we will encounter quantities represented by (

> You are given vectors

> Two vectors

> The scalar product of vectors

> Obtain a unit vector perpendicular to the two vectors given in Exercise 1.41. Exercise 1.41: Given two vectors

> Vector

> A cube is placed so that one corner is at the origin and three edges are along the x-, y-, and z-axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle between the edge along the z-axis (line ab) and the diagonal from the origin t

> Vectors

> In the methane molecule, CH4, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the C¬H bonds is in the direction of

> You are camping with Joe and Karl. Since all three of you like your privacy, you don’t pitch your tents close together. Joe’s tent is 21.0 m from yours, in the direction 23.0° south of east. Karl’s tent is 32.0 m from yours, in the direction 37.0° north

> Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 26.0 m in a direction 60.0° west of north. Jane walks 16.0 m in a direction 30.0° south of west. They the

> A dog in an open field runs 12.0 m east and then 28.0 m in a direction 50.0° west of north. In what direction and how far must the dog then run to end up 10.0 m south of her original starting point?

> A fence post is 52.0 m from where you are standing, in a direction 37.0° north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is 68.0 m?

> Which of the following are legitimate mathematical operations: (a)

> While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30.0° west of north, and finally walk 1.00 km at 32.0° north of east, where you find the treasure: a biography of Isaac Newto

> You decide to go to your favorite neighborhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km t

> A physical therapy patient has a forearm that weighs 20.5 N and lifts a 112.0-N weight. These two forces are directed vertically downward. The only other significant forces on this forearm come from the biceps muscle (which acts perpendicular to the fore

> A ship leaves the island of Guam and sails 285 km at 62.0° north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 km directly east of Guam?

> You are lost at night in a large, open field. Your GPS tells you that you are 122.0 m from your truck, in a direction 58.0° east of south. You walk 72.0 m due west along a ditch. How much farther, and in what direction, must you walk to reach your truck?

> An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at 60° north of west, and then 50 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and

> As a test of orienteering skills, your physics class holds a contest in a large, open field. Each contestant is told to travel 20.8 m due north from the starting point, then 38.0 m due east, and finally 18.0 m in the direction 33.0° west of south. After

> On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa, next to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. P1.66). The directions are shown relative to north: 0° is north, 90° is east, 1

> You leave the airport in College Station and fly 23.0 km in a direction 34.0o south of east. You then fly 46.0 km due north. How far and in what direction must you then fly to reach a private landing strip that is 32.0 km due west of the College Station

> A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, next 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. P1.64). Find the magn

> An object is thrown straight up into the air and feels no air resistance. How can the object have an acceleration when it has stopped moving at its highest point?

> A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.63. The pulls

> A plane leaves the airport in Galisteo and flies 170 km at 68.0° east of north; then it changes direction to fly 230 km at 36.0° south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in

> As noted in Exercise 1.26, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east of south, and then 280 m at 30° east of north. After a fourth displacement, she finds herself back where she started

> Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces

> Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 21.0° west of north, and the resultant of these two pulls is 460.0 N directly northward. Use vector components to find the magnitude of

> Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 72.0° between them. What pull does each rope exert if their resultant pull is 372 N directly upward?

> Estimate the number of atoms in your body.

> Biological tissues are typically made up of 98% water. Given that the density of water is 1.0 × 103 kg/m3, estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of 0.5 mm; (c) a honeybee.

> As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 ± 0.02 cm and a thickness of 0.050 ± 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume.

> A rectangular piece of aluminum is 7.60 ± 0.01 cm long and 1.90 ±0.01 cm wide. (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties

> You run due east at a constant speed of 3.00 m/s for a distance of 120.0 m and then continue running east at a constant speed of 5.00 m/s for another 120.0 m. For the total 240.0-m run, is your average velocity 4.00 m/s, greater than 4.00 m/s, or less th

> The black dots at the top of Fig. Q2.2 represent a series of high-speed photographs of an insect flying in a straight line from left to right (in the positive x-direction). Which of the graphs in Fig. Q2.2 most plausibly depicts this insectâ€&

> The density of air under standard laboratory conditions is 1.29 kg/m3, and about 20% of that air consists of oxygen. Typically, people breathe about 1/2 L of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is

> A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency

> In January 2006 astronomers reported the discovery of a planet, comparable in size to the earth, orbiting another star and having a mass about 5.5 times the earth’s mass. It is believed to consist of a mixture of rock and ice, similar to Neptune. If this

> An acre has a length of one furlong (1/8 mi) and a width one-tenth of its length. (a) How many acres are in a square mile? (b) How many square feet are in an acre? See Appendix E. (c) An acre-foot is the volume of water that would cover 1 acre of flat la

> Recall that density is mass divided by volume, and consult Appendix B as needed. (a) Calculate the average density of the earth in g/cm3, assuming our planet is a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end

> A movie stuntwoman drops from a helicopter that is 30.0 m above the ground and moving with a constant velocity whose components are 10.0 m/s upward and 15.0 m/s horizontal and toward the south. Ignore air resistance. (a) Where on the ground (relative to

> In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. For practice, a pilot drops a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 m above th

> Birds of prey typically rise upward on thermals. The paths these birds take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume that a bird completes a circle of radius 6.00 m ev

> A sly 1.5-kg monkey and a jungle veterinarian with a blow-gun loaded with a tranquilizer dart are 25 m above the ground in trees 70 m apart. Just as the veterinarian shoots horizontally at the monkey, the monkey drops from the tree in a vain attempt to e

> A bird flies in the xy-plane with a velocity vector given by v =(a - bt2)

> From the top of a tall building, you throw one ball straight up with speed v0 and one ball straight down with speed v0. (a) Which ball has the greater speed when it reaches the ground? (b) Which ball gets to the ground first? (c) Which ball has a greater

> A test rocket starting from rest at point A is launched by accelerating it along a 200.0-m incline at 1.90 m/s2 (Fig. P3.43). The incline rises at 35.0° above the horizontal, and at the instant the rocket leaves it, the engines turn off and th

> The position of a dragonfly that is flying parallel to the ground is given as a function of time by r = [2.90 m +(0.0900 m/s2)t2]

> If r = bt2

> A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket’s acceleration has components ax(t)= at2 and ay(t)= b - gt, where a = 2.50 m/s4, b = 9.00 m/s2, and g = 1.40 m/s3. At t = 0 the rocket is at the origi

> A rocket is fired at an angle from the top of a tower of height h0 = 50.0 m. Because of the design of the engines, its position coordinates are of the form x1t2= A + Bt2 and y(t) = C + Dt3, where A, B, C, and D are constants. The acceleration of the rock

> The velocity of blood in the aorta can be measured directly with ultrasound techniques. A typical graph of blood velocity versus time during a single heartbeat is shown in Fig. P2.92. Which statement is the best interpretation of this graph? (a) The bloo