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Many How Drops in All the Oceans?
Part A How many drops of water are in all the oceans on earth? Assume that contains 25 drops of water. Remember that this is an order-of-magnitude problem, so you should expect that you will only find rough estimates for the numbers you require. Hint A.1 Mean depth of the oceans The mean (average) depth of the oceans is about Hint A.2 Radius of the earth The radius of the earth is about . .
Hint A.3 Percent of the earth covered by ocean About 70% of the earth is covered by oceans. Hint A.4 Surface area of a sphere The surface area of a sphere with radius is given by the formula Express your answer to one significant figure. ANSWER: 4.01025 drops Correct .
Although order-of-magnitude calculations may seem silly at times, they are a major tool used by physicists. Any time that you are solving a problem in physics, it is helpful to have an estimate in your head of the order of magnitude that you expect from the answer. For instance, if you were trying to find the average speed of a car over a long trip and got an answer of 1000 miles per hour, you would immediately know that you had done something wrong, because your answer has the wrong order of magnitude. Order-of-magnitude problems are sometimes called Fermi problems, after the physicist Enrico Fermi who was reportedly a master of such approximate calculations. When the first atomic bomb was tested, Fermi was able to get a rough estimate of the power that the bomb released by throwing some torn bits of paper into the air as the pressure wave from the bomb passed him and then performing a rough calculation.
Tracking a Plane
A radar station, located at the origin of xz plane, as shown in the figure
, detects an airplane coming straight at the station from the east. At first observation (point A), the position of the airplane relative to the origin is . The position vector has a magnitude of 360 and is located at exactly 40
above the horizon. The airplane is tracked for another 123 in the vertical eastwest plane for 5.0 , until it has passed directly over the station and reached point B. The position of point B relative to the origin is (the magnitude of is 880 ). The contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward. Part A Define the displacement of the airplane while the radar was tracking it: What are the components of ? .
Hint A.1 How to approach the problem Keep in mind that subtraction, the x component of Hint A.2 .According to the rules of vector addition and is .
Finding the components of
What are the components of in the and directions? Express your answer in meters as an ordered pair, separating the x and z values with commas, to three significant figures. ANSWER: , = Answer not displayed Hint A.3 Finding the components of
Hint not displayed Express in meters as an ordered pair, separating the x and z components with a comma, to two significant figures. ANSWER: -1100,26 = Correct
Moving at the Speed of Light
Part A How many nanoseconds does it take light to travel a distance of 1.10 Hint A.1 How to approach the problem in vacuum?
Light travels at a constant speed; therefore, you can use the formula for the distance traveled in a certain amount of time by an object moving at constant speed. Before performing any calculations, it is often recommended, although it is not strictly necessary, to convert all quantities to their fundamental units rather than to multiples of the fundamental unit. Hint A.2 Find how many seconds it takes light to travel the given distance Given that the speed of light in vacuum is , how many seconds does it take
light to travel a distance of 1.10 ? Hint A.2.1 Find the time it takes light to travel a certain distance How long does it take light to travel a distance Hint A.2.1.1 The speed of an object ANSWER: ? Let be the speed of light.
Hint not displayed
Correct Hint A.2.2 Convert the given distance to meters Hint not displayed Express your answer numerically in seconds. ANSWER: Answer not displayed Express your answer numerically in nanoseconds.
ANSWER:
3670 Correct
Vector Addition Ranking Task
Six vectors ( through ) have the magnitudes and directions indicated in the figure.
Part A Rank the vector combinations on the basis of their magnitude. Hint A.1 Adding vectors graphically To add two vectors together, imagine sliding one vector (without rotating it) until its tail coincides with the tip of the second vector. The sum of the two vectors, termed the resultant vector , is the vector that goes from the tail of the first vector to the tip of the , is determined by the sum of the squares
second vector. The magnitude of the resultant, of its x and y components, that is,
Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:
View Correct Part B Rank the vector combinations on the basis of their angle, measured counterclockwise from
the positive x axis. Vectors parallel to the positive x axis have an angle of 0 . All angle measures fall between 0 and 360 . Hint B.1 Angle of a vector The angle of a vector is to be measured counterclockwise from the x axis, with the x
axis as 0 . The following vectors are at the angles listed and are shown on the graph below.
Notice that the magnitude of the vector is irrelevant when determining its angle Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:
View Correct
Adding Scalar Multiples of Vectors Graphically
Draw the vectors indicated. You may use any extra (unlabeled) vectors that are helpful; but, keep in mind that the unlabeled vectors should not be part of your submission.
Part A Draw the vector . Hint A.1 How to approach the problem You can add the vectors graphically or using components, but a graphical approach will be the simplest. It may help to draw the vector Hint A.2 Draw first.
Draw the vector . The length and orientation of the vector will be graded. The location of the vector is not important. ANSWER:
View Correct Drawing the vector may help you to find .
Hint A.3 Adding vectors graphically To add two vectors, slide one vector (without rotating it) until its tip coincides with the tail of the second vector. The sum of the two vectors is the vector that goes from the tail of the first vector to the tip of the second:
The length and orientation of the vector will be graded. The location of the vector is not important. ANSWER:
View Correct
Now use the same technique to answer the next two parts. Part B Draw the vector Hint B.1 Find . and
Draw the vectors and . Recall that multiplying a vector by a negative number reverses its direction. The length and orientation of the vectors will be graded. The locations of the vectors are not important. ANSWER:
View Correct Hint B.2 Adding vectors graphically To add two vectors, slide one vector (without rotating it) until its tip coincides with the tail of the second vector. The sum of the two vectors is the vector that goes from the tail of the first vector to the tip of the second:
The length and orientation of the vector will be graded. The location of the vector is not important. ANSWER:
View Correct Part C Draw the vector .
Hint C.1
Find
and Hint not displayed
Hint C.2 Adding vectors graphically Hint not displayed The length and orientation of the vector will be graded. The location of the vector is not important. ANSWER:
View Correct
Converting between Different Units
Unit conversion problems can seem tedious and unnecessary at times. However, different systems of units are used in different parts of the world, so when dealing with international transactions, documents, software, etc., unit conversions are often necessary. Here is a simple example. The inhabitants of a small island begin exporting beautiful cloth made from a rare plant that grows only on their island. Seeing how popular the small quantity that they export has been, they steadily raise their prices. A clothing maker from New York, thinking that he can save money by "cutting out the middleman," decides to travel to the small island and buy the cloth himself. Ignorant of the local custom of offering strangers outrageous prices and then negotiating down, the clothing maker accepts (much to everyone's surpise) the initial price of 400 Part A . The price of this cloth in New York is 120 .
If the clothing maker bought 500
of this fabric, how much money did he lose? Use
and . Hint A.1 How to approach the problem To find how much money the clothing maker loses, you must find how much money he spent and how much he would have spent in New York. Furthermore, since the problem asks how much he lost in dollars, you need to determine both in dollars. This will require unit conversions. Hint A.2 Find how much he paid If the clothing maker bought 500 at a cost of 400 , then simple multiplication will give how much he spent in tepizes. Once you've found that, convert to dollars. How much did the clothing maker spend in dollars?
Hint A.2.1 Find how much he paid in Hint not displayed Express your answer in dollars to three significant figures. ANSWER: 1.25105 Correct Hint A.3 Find the price in New York You know that the price of the fabric in New York is 120 . Thus, you need only to find the number of square yards that the clothing maker purchased and then multiply to find the price in New York. What would it have cost him to buy the fabric in New York? Hint A.3.1 Determine how much cloth he bought in Hint not displayed Express your answer in dollars to three significant figures. ANSWER: 7.18104 Correct Express your answer in dollars to three significant figures. ANSWER: 5.32104 Correct Still think that unit conversion isn't important? Here is a widely publicized, true story about how failing to convert units resulted in a huge loss. In 1998, the Mars Climate Orbiter probe crashed into the surface of Mars, instead of entering orbit. The resulting inquiry revealed that NASA navigators had been making minor course corrections in SI units, whereas the software written by the probe's makers implicitly used British units. In the United States, most scientists use SI units, whereas most engineers use the British, or Imperial, system of units. (Interestingly, British units are not used in Britain.) For these two groups to be able to communicate to one another, unit conversions are necessary. The unit of force in the SI system is the newton ( ), which is defined in terms of basic SI units as . Part B Find the value of 15.0 in pounds. Use the conversions and . The unit of force in the British system is the pound ( ), which is defined in terms of the slug (British unit of mass), foot ( ), and second ( ) as
. Hint B.1 How to approach the problem Hint not displayed Hint B.2 Calculate the first conversion Hint not displayed Express your answer in pounds to three significant figures. ANSWER: 3.37 15.0 = Correct Thus, if the NASA navigators believed that they were entering a force value of 15 (3.37
), they were actually entering a value nearly four and a half times higher, 15 . Though these errors were only in tiny course corrections, they added up during the trip of many millions of kilometers. In the end, the blame for the loss of the 125-million-dollar probe was placed on the lack of communication between people at NASA that allowed the units mismatch to go unnoticed. Nonetheless, this story makes apparent how important it is to carefully label the units used to measure a number.
Neptunium
In the fall of 2002, a group of scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about . The critical mass of a fissionable material is the minimum amount that must be brought together to start a chain reaction. Neptunium237 has a density of Part A .
What would be the radius of a sphere of neptunium-237 that has a critical mass? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Convert the critical mass to grams Hint not displayed Hint A.3 Find the needed volume Hint not displayed Hint A.4 Volume of a sphere Hint not displayed Express your answer in centimeters to three significant figures. ANSWER: = 9.02
Correct
Resolving Vector Components with Trigonometry
Often a vector is specified by a magnitude and a direction; for example, a rope with tension exerts a force of magnitude in a direction 35 north of east. This is a good way to think of vectors; however, to calculate results with vectors, it is best to select a coordinate system and manipulate the components of the vectors in that coordinate system. Part A Find the components of the vector with length to the x axis as shown. Hint A.1 What is the x component? = 1.00 and angle =20.0 with respect
Look at the figure shown.
points in the positive x direction, so
is positive. Also, the magnitude
is just the
length . Enter the x component followed by the y component, separated by a comma. ANSWER: 0.940,0.342 = Correct Part B Find the components of the vector with length to the x axis as shown. Hint B.1 What is the x component? = 1.00 and angle =20.0 with respect
The x component is still of the same form, that is, . Enter the x component followed by the y component, separated by a comma. ANSWER: 0.940,0.342 = Correct The components of still have the same form, that is, placement with respect to the y axis on the drawing. Part C Find the components of the vector with length = 1.00 and angle 25.0 as shown. Hint C.1 Method 1: Find the angle that makes with the positive x axis Hint not displayed Hint C.2 Method 2: Use vector addition Hint not displayed Enter the x component followed by the y component, separated by a comma. ANSWER: -0.423,0.906 = Correct , despite 's
Vector Dot Product
Let vectors , Calculate the following: Part A Hint A.1 , and .
Remember the dot product equation Hint not displayed
ANSWER: Part B
-10 = Correct
What is the angle between and ? Hint B.1 Remember the definition of dot products , where ANSWER: Part C ANSWER: Part D ANSWER: Part E Which of the following can be computed? Hint E.1 Dot product operator The dot product operates only on two vectors. The dot product of a vector and a scalar is not defined. ANSWER: 30 = Correct 30 = Correct 2.33 = Correct is the angle between and .
Correct and are different vectors with lengths Part F and respectively. Find the following:
Hint F.1 What is the angle between a vector and itself? The angle between a vector and itself is 0. Hint F.2 Remember the definition of dot products
, where
is the angle between
and
.
Express your answer in terms of ANSWER: = Correct Part G If and are perpendicular, Hint G.1 What is the angle between perpendicular vectors? The angle between vectors that are perpendicular is equal to ANSWER: = Part H If and are parallel, Hint H.1 What is the angle between parallel vectors? The angle between vectors that are parallel is equal to 0. Express your answer in terms of ANSWER: = Correct and . Correct radians or 90 degrees.
Vector Math Practice
Let vectors of the vectors along Part A , , , and , and , where are the components
respectively. Calculate the following:
Hint A.1 How to approach this problem Components can be multiplied by constants and added up individually. Express your answer as an ordered triplet of components the components. ANSWER: 14,2,15 = Correct Part B with commas to separate
Hint B.1 Magnitude of a vector Is the magnitude of a vector a scalar quantity or a vector quantity? Recall that a scalar quantity is described simply by a number. Express your answer as an ordered triplet magnitudes. ANSWER: 2.45,5.83,4.58 = Correct Part C ANSWER: Part D Determine the angle between and . Hint D.1 Definition of the dot product Hint not displayed Express your answer numerically in radians, to two significant figures. ANSWER: 1.8 = Correct radians Part E Express your answer as an ordered triplet of components the components. ANSWER: -20,11,12 = Correct Part F ANSWER: -39 = Correct with commas to separate 11 = Correct with commas to separate the
A Canoe on a River
A canoe has a velocity of 0.320 that is flowing at 0.510 southeast relative to the earth. The canoe is on a river
east relative to the earth.
Part A Find the magnitude of the velocity of the canoe relative to the river. Hint A.1 How to approach the problem In this problem there are two reference frames: the earth and the river. An observer standing on the edge of the river sees the canoe moving at 0.320 , whereas an observer Since the . drifting with the river current perceives the canoe as moving with velocity Note that the problem asks for the magnitude of Hint A.2 Find the relative velocity vector Let be the velocity of the canoe relative to the earth and the velocity of the water in .
velocity of the current in the river relative to the earth is known, you can determine
the river relative to the earth. What is the velocity Hint A.2.1 Relative velocity Consider a body A that moves with velocity
of the canoe relative to the river?
relative to a reference frame S and with relative
velocity relative to a second reference frame . If moves with speed to S, the velocity of the body relative to S is given by the vector sum . This equation is known as the Galilean transformation of velocity. ANSWER:
Correct Hint A.3 Find the components of the velocity of the canoe relative to the river Let the x axis point from west to east and the y axis from south to north. Find , the x and the y components of the velocity of the canoe relative to the river. Hint A.3.1 How to approach the problem The Galilean transformation of velocity tells you that the velocity of the canoe relative to the river is given by the difference of two vectors. Therefore, the components of the velocity of the canoe relative to the river are given by the difference of the components of those two vectors. Look back at the diagram from the introduction for help in setting up the equations. Hint A.3.2 Components of a vector Consider a vector components of that forms an angle are and where is the magnitude of the vector. Express the two velocity components, separated by a comma, in meters per second. ANSWER: -0.284,-0.226 , = Correct Now simply calculate the magnitude of , which is given by the square root of the sum of the squares of its components. Express your answer in meters per second. ANSWER: 0.363 = Correct Part B Find the direction the of velocity of the canoe relative to the river. Hint B.1 How to approach the problem The direction of a vector can be determined through simple trigonometric relations. You can use either the relation between the magnitude of the vector and one of its components or the relation between the two components of the vector. In both cases, use the information found in Part A. Note that the problem asks for the direction of south of west; your answer should be a positive angle between Hint B.2 Find the direction of a vector given its components as an angle measured and . with the positive x axis. The x and y and
Consider a vector of magnitude whose x component is is the angle this vector makes with the x axis? Hint B.2.1 The direction of a vector Consider a vector components are that forms an angle
and y component is
. What
with the positive x axis. The vector's x and y
and where is the magnitude of the vector. Thus,
, ANSWER:
, and
.
Correct Express your answer as an angle measured south of west. ANSWER: 38.6 degrees south of west Correct
A Wild Ride
A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by , where Part A At is the roller coaster car ascending or descending? Hint A.1 How to approach the problem is a positive dimensionless constant.
Hint not displayed Hint A.2 Find the vertical component of the velocity of the car Hint not displayed ANSWER: ascending descending Correct Part B Derive a general expression for the speed of the car. Hint B.1 How to approach the problem Hint not displayed Hint B.2 Magnitude of a vector Hint not displayed Hint B.3 Find the components of the velocity of the car Hint not displayed Express your answer in meters per second in terms of ANSWER: = Correct Part C The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding . Find the maximum value of Hint C.1 How to approach the problem Hint not displayed Hint C.2 Find the maximum value of the speed Hint not displayed Express your answer using two significant figures. ANSWER: 1.7 = Correct allowed by these regulations. and .
Circular Launch
A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 .
Part A How far from the bottom of the chute does the ball land? Hint A.1 Speed of ball upon leaving chute How fast is the ball moving at the top of the chute? Hint A.1.1 Equation of motion The centripetal acceleration for a particle moving in a circle is speed and is its instantaneous radius of rotation. ANSWER: = Correct Hint A.2 Time of free fall How long is the ball in free fall before it hits the ground? Hint A.2.1 Equation of motion Hint not displayed Hint A.2.2 Equation for the height of the ball Hint not displayed Express the free-fall time in terms of ANSWER: = Correct Hint A.3 Finding the horizontal distance The horizontal distance follows from found in Parts i and ii respectively. , where . and were and . , where is its
Your answer for the distance the ball travels from the end of the chute should contain ANSWER: = Correct
.
Crossing a River
A swimmer wants to cross a river, from point A to point B, as shown in the figure. The distance distance (from C to B) is 150 , and the speed (from A to C) is 200 , the . of the current in the river is 5
Suppose that the swimmer's velocity relative to the water makes an angle of with the line from A to C, as indicated in the figure. Part A To swim directly from A to B, what speed , relative to the water, should the swimmer have? Hint A.1 Use the motion in the y direction Suppose it takes the swimmer time to cross the river and arrive at B. Find an expression for by considering only (the y component of the swimmer's motion with respect to the shore). Hint A.1.1 The y component of the velocity relative to the shore Hint not displayed Answer in terms of ANSWER: , , and .
Correct Hint A.2 Use the motion in the x direction Now find an expression for by considering only the x component of the swimmer's motion. Hint A.2.1 The x component of the velocity relative to the shore Hint not displayed Express your answer in terms of ANSWER: , , , and .
Correct Note that this is the same time that you found when you considered only the motion in the y direction, even though the expressions are different. If you set both expressions equal to each other, you can solve for . Hint A.3 Solve for Find a symbolic expression for . If you used the previous hints, solve for eliminating from the two equations you derived. Express your answer in terms of , , , and ANSWER: = Answer not displayed . by
Express the swimmer's speed numerically, to three significant figures, in kilometers per hour. ANSWER: 4.04 = Correct Another way to do this problem is to add the swimmer's and the river's velocities vectorially, and set the angle that the resultant vector makes with AC or the river bank equal to that which AB makes with the same.
Curved Motion Diagram
The motion diagram shown in the figure represents a pendulum released from rest at an angle of 45 from the vertical. The dots in the motion diagram
represent the positions of the pendulum bob at eleven moments separated by equal time intervals. The green arrows represent the average velocity between adjacent dots. Also given is a "compass rose" in which directions are labeled with the letters of the alphabet. Part A What is the direction of the acceleration of the object at moment 5? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Definition of acceleration Hint not displayed Hint A.3 Change of velocity: a graphical interpretation Hint not displayed Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if the acceleration vector has zero length.
Correct Part B What is the direction of the acceleration of the object at moments 0 and 10? Hint B.1 Find the direction of the velocity What is the direction of the velocity of this object at moments 1 and 9? Enter the letters of the corresponding directions from the compass rose, separated by commas. Type Z if the velocity vector has zero length.
ANSWER:
directions at time step 1, time step 9 =
D,B Correct
Hint B.2 Definition of acceleration Acceleration is defined as the change in velocity per unit time. Mathematically,
. Since velocity is a vector, acceleration is a vector that points in the direction of the change in the velocity. Hint B.3 Applying the definition of acceleration To find the acceleration at moment 0, subtract the (vector) velocity at moment 0 from the velocity at moment 1. Similarly, to find the acceleration at moment 10, subtract the (vector) velocity at moment 9 from the velocity at moment 10. Enter the letters corresponding to the arrows with these directions from the compass rose in the figure, separated by commas. Type Z if the acceleration vector has zero length. ANSWER: D,F directions at time step 0, time step 10 = Correct
Arrow Hits Apple
An arrow is shot at an angle of distance Part A Find , the time that the arrow spends in the air. Hint A.1 Find the initial upward component of velocity in terms of D. Introduce the (unknown) variables and use kinematics to relate them and solve for initial velocity? Hint A.1.1 Find Hint A.1.2 Find Hint not displayed Express your answer symbolically in terms of and . for the initial components of velocity. Then . What is the vertical component of the above the horizontal. The arrow hits a tree a horizontal away, at the same height above the ground as it was shot. Use for the magnitude of the acceleration due to gravity.
Hint not displayed
ANSWER: = Correct Hint A.2 Find the time of flight in terms of the initial vertical component of velocity. From the change in the vertical component of velocity, you should be able to find terms of and . Hint A.2.1 Find When applied to the y-component of velocity, in this problem the formula for constant acceleration is in
with
What is , the vertical component of velocity when the arrow hits the tree? Answer symbolically in terms of only. ANSWER: = Correct Give your answer in terms of ANSWER: = Correct and .
Hint A.3 Put the algebra together to find
symbolically.
If you have an expression for the initial vertical velocity component in terms in terms of and , and another in terms of and , you should be able to eliminate this initial component to find an expression for Express your answer symbolically in terms of given variables. ANSWER: = Correct Answer numerically in seconds, to two significant figures. ANSWER: 6.7 = Correct Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. Part B
How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree? Hint B.1 When should the apple be dropped The apple should be dropped at the time equal to the total time it takes the arrow to reach the tree minus the time it takes the apple to fall 6.0 meters. Hint B.2 Find the time it takes for the apple to fall 6.0 meters How long does it take an apple to fall 6.0 meters? Express your answer numerically in seconds, to two significant figures. ANSWER: 1.1 = Correct Express your answer numerically in seconds, to two significant figures. ANSWER: 5.6 = Correct
The Archerfish
The archerfish is a type of fish well known for its ability to catch resting insects by spitting a jet of water at them. This spitting ability is enabled by the presence of a groove in the roof of the mouth of the archerfish. The groove forms a long, narrow tube when the fish places its tongue against it and propels drops of water along the tube by compressing its gill covers. When an archerfish is hunting, its body shape allows it to swim very close to the water surface and look upward without creating a disturbance. The fish can then bring the tip of its mouth close to the surface and shoot the drops of water at the insects resting on overhead vegetation or floating on the water surface. Part A At what speed should an archerfish spit the water to shoot down a floating insect located at a distance 0.800 from the fish? Assume that the fish is located very close to the surface of the pond and spits the water at an angle above the water surface. Hint A.1 How to approach the problem This problem involves projectile motion. A drop of water is launched at a certain angle above the surface, and you need to calculate the initial speed required to hit the target. Note that the target is an insect floating over the surface of the water. Hint A.2 Find how long it takes the water drop to fall back into the pond Assume that the drop is launched at an angle above the surface with an initial speed . The time it takes the drop to fall back into the water depends on the initial speed of the drop. Find a formula for to gravity. that depends only on , the angle and , the acceleration due
Hint A.2.1 Projectile motion The motion of a projectile can be described by two sets of equations: one that describes uniform motion at constant velocity in the horizontal direction and another one describing free-fall motion in the vertical direction. Specifically, the equations for displacement are
, where and are, respectively, the x coordinate and the y coordinate of the initial position of the projectile, and are, respectively, the x and y components of the initial velocity of the projectile, is the acceleration due to gravity, and is time. Note that if the launch angle by is known, then the components of the initial velocity are given
,
where
is the initial speed of the projectile.
Hint A.2.2 Initial and final position of the projectile Since the question asks for the time it takes the drop to fall back into the water, the initial and final vertical position of the drop must be the same. This gives you enough information to find from the y component of the equation for the projectile motion. Express your answer in terms of the angle due to gravity. ANSWER: = Correct , the initial speed , and , the acceleration
Hint A.3 Find how far from the fish the drop falls Now that you have found an expression for the time that it takes for the drop to fall back into the water in terms of the drop's initial speed , determine at what distance from the fish the drop falls. As previously, find an expression for in terms of using the information given in the problem introduction and your result found in the previous hint. Express your answer in terms of the angle due to gravity. ANSWER: = Correct , the initial speed , and , the acceleration
Now make the distance equal to the location of the insect and solve for Express your answer in meters per second. ANSWER: 3.01 = Correct
.
Some archerfish can "shoot" as far as 3.5 , hitting their targets with reasonable accuracy as far as 1.2 . They have binocular vision, which helps them judge distance. Part B Now assume that the insect, instead of floating on the surface, is resting on a leaf above the water surface at a horizontal distance 0.600 away from the fish. The archerfish successfully shoots down the resting insect by spitting water drops at the same angle above the surface and with the same initial speed surface was the insect? Hint B.1 How to approach the problem as before. At what height above the
Use the kinematics equations that describe projectile motion. However, now you are given the projectile's initial speed, launch angle, and final distance along the x axis and you need to calculate the corresponding height reached by the projectile. Hint B.2 Find the time it takes the water drop to hit the insect Considering that a water drop launched by the archerfish successfully hit the insect resting on the leaf, how long did it take? Hint B.2.1 Kinematics equation in the horizontal direction Hint not displayed Express your answer in seconds. ANSWER: 0.399 = Correct Now calculate the height reached by the drop at this time. Express your answer in meters. ANSWER: 0.260 = Correct Experiments have shown that the archerfish can predict the point where the disabled prey will fall and hit the water by simply looking at the initial trajectory of the dislodged insect for only 10 . The archerfish then darts to the place where it has "calculated" the insect will hit the water, planning to get there before another fish does.
Delivering a Package by Air
A relief airplane is delivering a food package to a group of people stranded on a very small island. The island is too small for the plane to land on, and the only way to deliver the package is by dropping it. The airplane flies horizontally with constant speed of 240 at an altitude of 750 . The positive x and y directions are defined in the figure. For all parts, assume that the "island" refers to the point at a distance from the point at which the package is released, as shown in the figure. Ignore the height of this point above sea level. Assume that the acceleration due to gravity is = 9.80 Part A .
After a package is ejected from the plane, how long will it take for it to reach sea level from the time it is ejected? Assume that the package, like the plane, has an initial velocity of 240 in the horizontal direction. Hint A.1 Knowns and unknowns: what are the initial conditions? Take the origin of the coordinate system to be at the point on the surface of the water directly below the point at which the package is released. The directions of the axes are shown in the figure in the problem introduction. In this coordinate system, what are the values of , , , of the package? Hint A.1.1 Initial velocity in the y direction Hint not displayed Express your answers numerically and enter them, separated by commas, in the order , , . Use units of meters and mph for distances and speeds, respectively. ANSWER: 0,750,240,0 ,, , = Correct Hint A.2 What are the knowns and unknowns when the package hits the ground? Take the origin of the coordinate system to be at the point on the surface of the water directly below the point at which the package is released. The directions of the axes are shown in the figure in the problem introduction. Let be the time when the package hits ,
the ground. In this coordinate system, 240 since there is no acceleration in the x direction. Which of the following values is/are known? Check all that apply. ANSWER:
Correct Of course, you also know that .
Hint A.3 Find the best equation to use Which of the equations below could you use to find the time ground? Hint A.3.1 How to determine which equation to use Hint not displayed ANSWER: when the packet hits the
Correct Substitute the values into this equation and solve for . Express your answer numerically in seconds. Neglect air resistance. ANSWER: 12.4 = Correct Part B If the package is to land right on the island, at what horizontal distance the island should the package be released? Hint B.1 How to approach the problem from the plane to
You are asked to find , which is also the change in the x coordinate of the package over the time spent in the air. You should have calculated this time interval in Part A. Use it to find . The equation for is Hint B.2
Since there is no acceleration in the horizontal direction, the equation for
. Express the distance numerically in meters. ANSWER: 1330 = Correct Part C What is the speed of the package when it hits the ground? Hint C.1 How to approach the problem Hint not displayed Hint C.2 The equation for the velocity in the y direction Hint not displayed Express your answer numerically in miles per hour. ANSWER: 362 = Correct Part D The speed at which the package hits the ground is really fast! If a package hits the ground at such a speed, it can be crushed and also cause some serious damage on the ground. Which of the following would help decrease the speed with which the package hits the ground? ANSWER: Increase the plane's speed and height Decrease the plane's speed and height Correct This is why it would be nice for rescue teams to have hybrid airplane-helicopters. Of course, then
they can just airlift the stranded group.

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UC Davis - ENG - 102

UC Davis - ENG - 102

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UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

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UC Davis - ENG - 102

UC Davis - ENG - 102

UC Davis - ENG - 102

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UC Davis - ENG - 102

UC Davis - ENG - 102

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UC Davis - ENG - 102

UC Davis - ENG - 102

Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of thi

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UC Davis - ENG - 102

UC Davis - ENG - 102

MT (1) 1696!2200 pixels3/3/09 11:17 AMfile:/Users/raissa/Documents/Classes/Eng102-W09/Exams/mt/MT%20(1)Page 1 of 1MT (2) 1696!2200 pixels3/3/09 11:18 AMfile:/Users/raissa/Documents/Classes/Eng102-W09/Exams/mt/MT%20(2)Page 1 of 1MT (3) 1696!2200 pi

UC Davis - ENG - 102

UC Davis - ENG - 104

UC Davis - ENG - 104

UC Davis - ENG - 104