Assignment - Chapter 4 Due: 11:59pm on Thursday, February 4, 2010 Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy [Return to Standard Assignment View ] An Object Accelerating on a Ramp Learning Goal: Understand that the acceleration vector is in the direction of the change of the velocity vector. In one dimensional (straight line) motion, acceleration is accompanied by a change in speed, and the acceleration is always parallel (or antiparallel) to the velocity. When motion can occur in two dimensions (e.g. is confined to a tabletop but can lie anywhere in the x-y plane), the definition of acceleration is in the limit . In picturing this vector derivative you can think of the derivative of a vector as an instantaneous quantity by thinking of the velocity of the tip of the arrow as the vector changes in time. Alternatively, you can (for small ) approximate the acceleration as . Obviously the difference between and is another vector that can lie in any direction. If it is longer but in the same direction, will be parallel to . On the other hand, if has the same magnitude as but is in a slightly different direction, then will be perpendicular to . In general, can differ from in both magnitude and direction, hence can have any direction relative to . This problem contains several examples of this.Consider an object sliding on a frictionless ramp as depicted here. The object is already moving along the ramp toward position 2 when it is at position 1. The following questions concern the direction of the object's acceleration vector, . In this problem, you should find the direction of the acceleration vector by drawing the velocity vector at two points near to the position you are asked about. Note that since the object moves along the track, its velocity vector at a point will be tangent to the track at that point. The acceleration vector will point in the same direction as the vector difference of the two velocities. (This is a result of the equation given above.) Part A Which direction best approximates the direction of when the object is at position 1? Hint A.1 Consider the change in velocity Hint not displayed ANSWER: straight up downward to the left downward to the right straight down Correct Part B Which direction best approximates the direction of when the object is at position 2? Hint B.1 Consider the change in velocity Hint not displayed ANSWER: straight up upward to the right straight down downward to the left Correct Even though the acceleration is directed straight up, this does not mean that the object is moving straight up. Part C Which direction best approximates the direction of when the object is at position 3? Hint C.1 Consider the change in velocity Hint not displayed ANSWER: upward to the right MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentID... 1 of 19 4/22/2010 5:17 PM
Transcript
Assignment - Chapter 4
Due: 11:59pm on Thursday, February 4, 2010
Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy
[Return to Standard Assignment View]
An Object Accelerating on a Ramp
Learning Goal: Understand that the acceleration vector is in the direction of the change of the velocity vector.
In one dimensional (straight line) motion, acceleration is accompanied by a change in speed, and the acceleration is always parallel (or antiparallel) to the velocity.
When motion can occur in two dimensions (e.g. is confined to a tabletop but can lie anywhere in the x-y plane), the definition of acceleration is
in the limit .
In picturing this vector derivative you can think of the derivative of a vector as an instantaneous quantity by thinking of the velocity of the tip of the arrow as the vector changes in time. Alternatively, you can (for
small ) approximate the acceleration as
.
Obviously the difference between and is another vector that can lie in any direction. If it is longer but in the same direction, will be parallel to . On the other hand, if has the
same magnitude as but is in a slightly different direction, then will be perpendicular to . In general, can differ from in both magnitude and direction, hence can have any
direction relative to .
This problem contains several examples of this.Consider an object sliding on a frictionless ramp as depicted here. The object is already movingalong the ramp toward position 2 when it is at position 1. The following questions concern the direction of the object's acceleration vector, . In this
problem, you should find the direction of the acceleration vector by drawing the velocity vector at two points near to the position you are askedabout. Note that since the object moves along the track, its velocity vector at a point will be tangent to the track at that point. The acceleration vector
will point in the same direction as the vector difference of the two velocities. (This is a result of the equation given
above.)
Part A
Which direction best approximates the direction of when the object is at position 1?
Hint A.1 Consider the change in velocity
Hint not displayed
ANSWER: straight up
downward to the left
downward to the right
straight down
Correct
Part B
Which direction best approximates the direction of when the object is at position 2?
Hint B.1 Consider the change in velocity
Hint not displayed
ANSWER: straight up
upward to the right
straight down
downward to the left
Correct
Even though the acceleration is directed straight up, this does not mean that the object is moving straight up.
Part C
Which direction best approximates the direction of when the object is at position 3?
Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently.
Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis weassume that air resistance can be neglected.
An object undergoing projectile motion near the surface of the earth obeys the following rules:
An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, , is constant.1.
An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by , is equal to 9.80 near the surface of the earth. Hence, the y
component of its velocity, , changes continuously.
2.
An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though thehorizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time the projectile is in the air.
3.
The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time corresponds to the
moment just after the ball is launched from position and . Its launch velocity, also called the initial velocity, is .
Two other points along the trajectory are indicated in the figure.
One is the moment the ball reaches the peak of its trajectory, at time with velocity . Its position at this moment is denoted by
or since it is at its maximum height.
The other point, at time with velocity , corresponds to the moment just before the ball strikes the ground on the way back down. At this
time its position is , also known as ( since it is at its maximum horizontal range.
Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case
here. Hence .
Part A
How do the speeds , , and (at times , , and ) compare?
ANSWER: = = > 0
= > = 0
= > > 0
> > > 0
> > = 0
Correct
Here equals by symmetry and both exceed . This is because and include vertical speed as well as the constant horizontal speed.
Consider a diagram of the ball at time . Recall that refers to the instant just after the ball has been launched, so it is still at ground level
( ). However, it is already moving with initial velocity , whose magnitude is and direction is
counterclockwise from the positive x direction.
Part B
What are the values of the intial velocity vector components and (both in ) as well as the acceleration vector components and (both in )? Here the subscript 0 means "at time ."
Hint B.1 Determining components of a vector that is aligned with an axis
Hint not displayed
Hint B.2 Calculating the components of the initial velocity
Hint not displayed
ANSWER:
Correct
Also notice that at time , just before the ball lands, its velocity components are (the same as always) and (the same size but opposite sign from by symmetry).
The acceleration at time will have components (0, -9.80 ), exactly the same as at , as required by Rule 2.
The peak of the trajectory occurs at time . This is the point where the ball reaches its maximum height . At the peak the ball switches from moving up to moving down, even as it continues to travel
horizontally at a constant rate.
Part C
What are the values of the velocity vector components and (both in ) as well as the acceleration vector components and (both in )? Here the subscript 1 means that these are all at
time .
ANSWER:
Correct
At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directedchange in velocity, so the ball is accelerating downward even at the peak.
The flight time refers to the total amount of time the ball is in the air, from just after it is launched ( ) until just before it lands ( ). Hence the flight time can be calculated as , or just in this
particular situation since . Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down.
The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air.
Part D
If a second ball were dropped from rest from height , how long would it take to reach the ground? Ignore air resistance.
Hint D.1 Kicking a ball of cliff; a related problem
Hint not displayed
Check all that apply.
ANSWER:
Correct
In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground.
The range of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as , or just in this particular situation since .
Range can be calculated as the product of the flight time and the x component of the velocity (which is the same at all times, so ). The value of can be found from the launch speed and the
launch angle using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from and using trig functions.
The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position ( ) at time with initial
speed and launch angle measured from the horizontal. As was the case above, refers to the flight time and refers to the range of the projectile.
flight time:
range:
In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range.The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees.
Part E
Which of the following changes would increase the range of the ball shown in the original figure?
Check all that apply.
ANSWER: Increase above 30 .
Reduce below 30 .
Reduce from 60 to 45 .
Reduce from 60 to less than 30 .
Increase from 60 up toward 90 .
Correct
A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object doesnot land at the same height from which is was launched, the rules given in the introduction will still be useful.
Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influencedby air. The value of the gravitational free-fall acceleration is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers,
like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable.
Learning Goal: To understand the basic concepts of projectile motion.
Projectile motion may seem rather complex at first. However, by breaking it down into components, you will find that it is really no different than the one-dimensional motions that you have already studied.
One of the most often used techniques in physics is to divide two- and three-dimensional quantities into components. For instance, in projectile motion, a particle has some initial velocity . In general, this
velocity can point in any direction on the xy plane and can have any magnitude. To make a problem more managable, it is common to break up such a quantity into its x component and its y component .
Consider a particle with initial velocity that has magnitude 12.0 and is directed 60.0 above the negative x axis.
Part A
What is the x component of ?
Express your answer in meters per second.
ANSWER: = -6.00
Correct
Part B
What is the y component of ?
Express your answer in meters per second.
ANSWER: =
10.4
Correct
Breaking up the velocities into components is particularly useful when the components do not affect each other. Eventually, you will learn about situations in which the components of velocity do affect oneanother, but for now you will only be looking at problems where they do not. So, if there is acceleration in the x direction but not in the y direction, then the x component of the velocity will change, but the ycomponent of the velocity will not.
Part C
Look at this applet. The motion diagram for a projectile is displayed, as are the motion diagrams for each component. The x-component motion diagram is what you would get if you shined a spotlight down on
the particle as it moved and recorded the motion of its shadow. Similarly, if you shined a spotlight to the left and recorded the particle's shadow, you would get the motion diagram for its y component. Howwould you describe the two motion diagrams for the components?
ANSWER: Both the vertical and horizontal components exhibit motion with constant nonzero acceleration.
The vertical component exhibits motion with constant nonzero acceleration, whereas the horizontal component exhibits constant-velocity motion.
The vertical component exhibits constant-velocity motion, whereas the horizontal component exhibits motion with constant nonzero acceleration.
Both the vertical and horizontal components exhibit motion with constant velocity.
Correct
As you can see, the two components of the motion obey their own independent kinematic laws. For the vertical component, there is an acceleration downward with magnitude . Thus, you
can calculate the vertical position of the particle at any time using the standard kinematic equation . Similarly, there is no acceleration in the horizontal direction, so the
horizontal position of the particle is given by the standard kinematic equation .
Now, consider this applet. Two balls are simultaneously dropped from a height of 5.0 .
Part D
How long does it take for the balls to reach the ground? Use 10 for the magnitude of the acceleration due to gravity.
Hint D.1 How to approach the problem
Hint not displayed
Express your answer in seconds to two significant figures.
ANSWER: = 1.0
Correct
This situation, which you have dealt with before (motion under the constant acceleration of gravity), is actually a special case of projectile motion. Think of this as projectile motion where the horizontalcomponent of the initial velocity is zero.
Part E
Imagine the ball on the left is given a nonzero initial speed in the horizontal direction, while the ball on the right continues to fall with zero initial velocity. What horizontal speed must the ball on the left
start with so that it hits the ground at the same position as the ball on the right?
Hint E.1 How to approach the problem
Hint not displayed
Express your answer in meters per second to two significant figures.
ANSWER: = 3.0
Correct
You can adjust the horizontal speeds in this applet. Notice that regardless of what horizontal speeds you give to the balls, they continue to move vertically in the same way (i.e., they are at the same ycoordinate at the same time).
Learning Goal: To practice Problem-Solving Strategy 4.1 for projectile motion problems.
A rock thrown with speed 6.50 and launch angle 30.0 (above the horizontal) travels a horizontal distance of = 12.0 before hitting the ground. From what height was the rock thrown? Use the value
MODEL: Make simplifying assumptions, such as treating the object as a particle. Is it reasonable to ignore air resistance?
VISUALIZE: Use a pictorial representation. Establish a coordinate system with the x axis horizontal and the y axis vertical. Show important points in the motion on a sketch. Define symbols, and identify whatyou are trying to find.
SOLVE: The acceleration is known: and . Thus, the problem becomes one of two-dimensional kinematics. The kinematic equations are
,
.
is the same for the horizontal and vertical components of the motion. Find from one component, and then use that value for the other component.
ASSESS: Check that your result has the correct units, is reasonable, and answers the question.
Model
Start by making simplifying assumptions: Model the rock as a particle in free fall. You can ignore air resistance because the rock is a relatively heavy object moving relatively slowly.
Visualize
Part A
Which diagram represents an accurate sketch of the rock's trajectory?
Hint A.1 The launch angle
Hint not displayed
ANSWER:
Correct
Part B
As stated in the strategy, choose a coordinate system where the x axis is horizontal and the y axis is vertical. Note that in the strategy, the y component of the projectile's acceleration, , is taken to be
negative. This implies that the positive y axis is upward. Use the same convention for your y axis, and take the positive x axis to be to the right. Where is the best location to place the origin of the coordinatesystem?
At ground level below the point where the rock is launched
At ground level below the peak of the trajectory
Correct
It's best to place the origin of the coordinate system at ground level below the launching point because in this way all the points of interest (the launching point and the landing point) will have positivecoordinates. (Based on your experience, you know that it's generally easier to work with positive coordinates.) Keep in mind, however, that this is an arbitrary choice. The correct solution of theproblem will not depend on the location of the origin of your coordinate system.
Now, define symbols representing initial and final position, velocity, and time. Your target variable is , the initial y coordinate of the rock. Your pictorial representation should be complete now, and
similar to the picture below:
Solve
Part C
Find the height from which the rock was launched.
Hint C.1 How to approach the problem
Hint not displayed
Hint C.2 Find the time spent in the air
Hint not displayed
Hint C.3 Find the y component of the initial velocity
Hint not displayed
Express your answer in meters to three significant figures.
ANSWER: = 15.4
Correct
Assess
Part D
A second rock is thrown straight upward with a speed 3.250 . If this rock takes 2.132 to fall to the ground, from what height was it released?
Hint D.1 Identify the known variables
Hint not displayed
Hint D.2 Determine which equation to use to find the height
Hint not displayed
Express your answer in meters to three significant figures.
ANSWER: = 15.4
Correct
Projectile motion is made up of two independent motions: uniform motion at constant velocity in the horizontal direction and free-fall motion in the vertical direction. Because both rocks were thrown
with the same initial vertical velocity, 3.250 , and fell the same vertical distance of 15.4 , they were in the air for the same amount of time. This result was expected and helps to confirm
that you did the calculation in Part C correctly.
Battleship Shells
A battleship simultaneously fires two shells toward two identical enemy ships. One shell hits ship A, which is close by, and the other hits ship B, which is farther away. The two shells are fired at the samespeed. Assume that air resistance is negligible and that the magnitude of the acceleration due to gravity is .
Part A
What shape is the trajectory (graph of y vs. x) of the shells?
For two shells fired at the same speed which statement about the horizontal distance traveled is correct?
Hint B.1 Two things to consider
Hint not displayed
ANSWER: The shell fired at a larger angle with respect to the horizontal lands farther away.
The shell fired at an angle closest to 45 degrees lands farther away.
The shell fired at a smaller angle with respect to the horizontal lands farther away.
The lighter shell lands farther away.
Correct
Consider the situation in which both shells are fired at an angle greater than 45 degrees with respect to the horizontal. Remember that enemy ship A is closer than enemy ship B.
Part C
Which shell is fired at the larger angle?
Hint C.1 Consider the limiting case
Hint not displayed
ANSWER: A
B
Both shells are fired at the same angle.
Correct
Part D
Which shell is launched with a greater vertical velocity, ?
ANSWER: A
B
Both shells are launched with the same vertical velocity.
Correct
Part E
Which shell is launched with a greater horizontal velocity, ?
ANSWER: A
B
Both shells are launched with the same horizontal velocity.
Correct
Part F
Which shell reaches the greater maximum height?
Hint F.1 What determines maximum height?
Hint not displayed
ANSWER: A
B
Both shells reach the same maximum height.
Correct
Part G
Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)?
A cannon is fired from the top of a cliff as shown in the figure. Ignore drag (air friction) for this question. Take as the height of the cliff.
Part A
Which of the paths would the cannonball most likely follow if the cannon barrel is horizontal?
Hint A.1 Find the y position as a function of time
Hint not displayed
Hint A.2 Interpreting your equation
Hint not displayed
ANSWER: A
B
C
D
Correct
Part B
Now the cannon is pointed straight up and fired. (This procedure is not recommended!) Under the conditions already stated (drag is to be ignored) which of the following correctly describes the acceleration ofthe ball?
ANSWER: A steadily increasing downward acceleration from the moment the cannonball leaves the cannon barrel until it reaches its highest point
A steadily decreasing upward acceleration from the moment the cannonball leaves the cannon barrel until it reaches its highest point
A constant upward acceleration
A constant downward acceleration
Correct
The acceleration of the cannonball after it is fired is the constant acceleration due to gravity.
Speed of a Bullet
A bullet is shot through two cardboard disks attached a distance apart to a shaft turning with a rotational period , as shown.
Part A
Derive a formula for the bullet speed in terms of , , and a measured angle between the position of the hole in the first disk and that of the hole in the second. If required, use , not its numeric
equivalent. Both of the holes lie at the same radial distance from the shaft. measures the angular displacement between the two holes; for instance, means that the holes are in a line and means
that when one hole is up, the other is down. Assume that the bullet must travel through the set of disks within a single revolution.
Hint A.1 Consider hole positions
The relative position of the holes can be used to find the bullet's speed. Remember, the shaft will have rotated while the bullet travels between the disks.
Hint A.2 How long does it take for the disks to rotate by an angle ?
The disks rotate by 2 in time . How long will it take them to rotate by ?
Hint A.2.1 Checking your formula
Hint not displayed
Give your answer in terms of , , and constants such as .
ANSWER:
=
Correct
You know that the bullet went a distance in the time it took for the disks to rotate by .
ANSWER:
=
Correct
Direction of Velocity at Various Times in Flight for Projectile Motion Conceptual Question
For each of the motions described below, determine the algebraic sign (positive, negative, or zero) of the x component and y component of velocity of the object at the time specified. For all of the motions, the
positive x axis points to the right and the positive y axis points upward.
Alex, a mountaineer, must leap across a wide crevasse. The other side of the crevasse is below the point from which he leaps, as shown in thefigure. Alex leaps horizontally and successfully makes the jump.
Part A
Determine the algebraic sign of Alex's x velocity and y velocity at the instant he leaves the ground at the beginning of the jump.
Hint A.1 Algebraic sign of velocity
Hint not displayed
Hint A.2 Sketch Alex's initial velocity
Hint not displayed
Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+).
ANSWER: +,0
Correct
Part B
Determine the algebraic signs of Alex's x velocity and y velocity the instant before he lands at the end of the jump.
Hint B.1 Sketch Alex's final velocity
Hint not displayed
Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+).
ANSWER: +,-
Correct
At the buzzer, a basketball player shoots a desperation shot. The ball goes in!
Determine the algebraic signs of the ball's x velocity and y velocity the instant after it leaves the player's hands.
Hint C.1 Sketch the basketball's initial velocity
Hint not displayed
Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+).
ANSWER: +,+
Correct
Part D
Determine the algebraic signs of the ball's x velocity and y velocity at the ball's maximum height.
Hint D.1 Sketch the basketball's velocity at maximum height
Hint not displayed
Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+).
ANSWER: +,0
Correct
Throwing Stones
Two stones are launched from the top of a tall building. One stone is thrown in a direction 25.0 above the horizontal with a speed of 20.0 ; the other is thrown in a direction 25.0 below the horizontal
with the same speed.
Part A
Which stone spends more time in the air? (Neglect air resistance.)
Hint A.1 Determine the stones' trajectories
Hint not displayed
Hint A.2 Compare the speed of the stones
Hint not displayed
ANSWER: The stone thrown upward spends more time in the air.
The stone thrown downward spends more time in the air.
Both stones spend the same amount of time in the air.
Correct
When the stone thrown upward returns to the level of the top of the building, it is moving at 20.0 in a direction 25.0 below the horizontal. This is the initial speed of the stone that is thrown
downward. Since the stone that is thrown upward must reach the top of its trajectory before returning to a height that is level with the top of the building, it takes the stone thrown upward more time to
strike the ground.
Part B
Which stone lands farther away from the building? (Neglect air resistance.)
Hint B.1 Compare the horizontal speed of the stones
Hint not displayed
ANSWER: The stone thrown upward lands farther from the building.
The stone thrown downward lands farther from the building.
Both stones land the same distance from the building.
Correct
Both stones have the same horizontal speed. The horizontal distance traveled by one of the stones is given by the product of the x component of its velocity and the time the stone spends in the air. FromPart A, the time spent in the air by the stone that is thrown upward is greater than that spent in the air by the stone thrown downward. Therefore, the stone that is thrown upward travels farther in thehorizontal direction.
Conceptual Question 4.1
Part A
Is the particle in the figure speeding up, slowing down, or traveling at constant speed?
A sailor climbs to the top of the mast, above the deck, to look for land while his ship moves steadily forward through calm waters at . Unfortunately, he drops his spyglass to the deck below.
Part A
Where does it land with respect to the base of the mast below him?
ANSWER: near the base of the mast
7.0 meters in front of the base of the mast
7.0 meters behind of the base of the mast
None of the above
Correct
Part B
Where does it land with respect to a fisherman sitting at rest in his dinghy as the ship goes past? Assume that the fisherman is even with the mast at the instant the spyglass is dropped.
ANSWER: 7.00
Correct m
Problem 4.21
The figure shows the angular-velocity-versus-time graph for a particle moving in a circle.
Part A
How many revolutions does the object make during the first 4 s?
How fast must a plane fly along the earth's equator so that the sun stands still relative to the passengers? Give your answer in both km/hr and mph.
Part A
ANSWER: 1680
Correct
Part B
ANSWER: 1040
Correct mph
Part C
In which direction must the plane fly, east to west or west to east?
ANSWER: east to west
west to east
Correct
Problem 4.56
A kayaker needs to paddle north across a 100-m-wide harbor. The tide is going out, creating a tidal current that flows to the east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 m/s.
Part A
In which direction should he paddle in order to travel straight across the harbor?
ANSWER: west of north
west of north
west of north
west of north
Correct
Part B
How long will it take him to cross?
ANSWER: 44.7
Correct s
Reading Quiz 4.3
Part A
The quantity with the symbol is called
ANSWER: the circular weight.
the angular velocity.
the circular velocity
the centripetal acceleration.
Correct
Reading Quiz 4.2
Part A
A hunter points a rifle directly at a coconut that he intends to shoot off a tree. It so happens that the coconut falls from the tree at the exact instant the hunter pulls the trigger. What happens to the bullet?
A ball is thrown upward at a 45 angle. In the absence of air resistance, the ball follows a
ANSWER: tangential curve.
parabolic curve.
sine curve.
linear curve.
Correct
Reading Quiz 4.4
Part A
For uniform circular motion, the acceleration
ANSWER: points toward the center of the circle.
points toward the outside of the circle.
is tangent to the circle.
is zero.
Correct
True/False Problem 4.1
Part A
A ball is thrown at an angle above the horizontal from the top of a cliff and feels no air resistance. A runner at the base of the cliff moves horizontally so that she is always under the ball. In this runner'sreference frame, the path of the ball is a straight line rather than a parabola.
ANSWER: True
False
Correct
True/False Problem 4.2
Part A
A tennis ball following a parabolic trajectory without air resistance has two forces acting on it; gravity downward and a force keeping it moving forward.
ANSWER: True
False
Correct
True/False Problem 4.3
Part A
A grasshopper leaps into the air at a 62 degree angle above the horizontal. At its highest point, the grasshopper's velocity and acceleration are equal to zero.
ANSWER: True
False
Correct
True/False Problem 4.6
Part A
Two identical projectiles are launched with equal speeds from the top of a building and feel no air resistance. Projectile is launched above the horizontal while projectile is launched below the
horizontal. Both projectiles have exactly the same acceleration while they are in the air.
Satellites in orbit are accelerated toward the earth, so they must be getting closer and closer to our planet.
ANSWER: True
False
Correct
True/False Problem 4.10
Part A
Satellites in circular orbits around the earth are in equilibrium because they have uniform speed.
ANSWER: True
False
Correct
Advice for the Quarterback
A quarterback is set up to throw the football to a receiver who is running with a constant velocity directly away from the quarterback and is now a distance away from the quarterback. The quarterback
figures that the ball must be thrown at an angle to the horizontal and he estimates that the receiver must catch the ball a time interval after it is thrown to avoid having opposition players prevent the receiver
from making the catch. In the following you may assume that the ball is thrown and caught at the same height above the level playing field. Assume that the y coordinate of the ball at the instant it is thrown orcaught is and that the horizontal position of the quaterback is .
Use for the magnitude of the acceleration due to gravity, and use the pictured inertial coordinate system when solving the problem.
Part A
Find , the vertical component of the velocity of the ball when the quarterback releases it.
Hint A.1 Equation of motion in y direction
Hint not displayed
Hint A.2 Height at which the ball is caught,
Hint not displayed
Express in terms of and .
ANSWER:
=
Correct
Part B
Find , the initial horizontal component of velocity of the ball.
Find the speed with which the quarterback must throw the ball.
Hint C.1 How to approach the problem
Hint not displayed
Answer in terms of , , , and .
ANSWER:
=
Correct
Part D
Assuming that the quarterback throws the ball with speed , find the angle above the horizontal at which he should throw it.
Hint D.1 Find angle from and
Hint not displayed
Your solution should contain an inverse trig function (entered as asin, acos, or atan). Give your answer in terms of already known quantities, , , and .
ANSWER:
=
Correct
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 300 at an altitude of 600 .
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 300 in the horizontal
direction.
Hint A.1 Knowns and unknowns: what are the initial conditions?
Hint not displayed
Hint A.2 What are the knowns and unknowns when the package hits the ground?
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Hint A.3 Find the best equation to use
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Express your answer numerically in seconds. Neglect air resistance.
ANSWER: = 11.1
Correct
Part B
If the package is to land right on the island, at what horizontal distance from the plane to the island should the package be released?
What is the speed of the package when it hits the ground?
Hint C.1 How to approach the problem
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Hint C.2 The equation for the velocity in the y direction
Hint not displayed
Express your answer numerically in miles per hour.
ANSWER: =
386
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.
Problem 4.19
Susan, driving north at , and Shawn, driving east at , are approaching an intersection.
Part A
What is Shawn's speed relative to Susan's reference frame?
ANSWER: 75.0
Correct mph
Score Summary:
Your score on this assignment is 95.5%.
You received 30.55 out of a possible total of 32 points.
