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PHCC 141: Physics for Scientists and Engineers I - Fall 2007
3a. Motion in Two or Three Dimensions
Due at 11:59pm on Friday, September 7, 2007
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Number of answer attempts per question is: 5 You gain credit for: correctly answering a question in a Part, or correctly answering a question in a Hint. You lose credit for: exhausting all attempts or requesting the answer to a question in a Part or Hint, or incorrectly answering a question in a Part. Late submissions: reduce your score by 100% over each day late. Hints are helpful clues or simpler questions that guide you to the answer. Hints are not available for all questions. There is no penalty for leaving questions in Hints unanswered. Grading of Incorrect Answers
For Multiple-Choice or True/False questions, you lose 100%/(# of options - 1) credit per incorrect answer. For any other question, you lose 3% credit per incorrect answer. Standard kinematic definitions, two body problems
Position, Velocity, and Acceleration
Learning Goal: To identify situations when position, velocity, and /or acceleration change, realizing that change can be in direction or magnitude. (measured from a nonaccelerating reference frame), then the object's If an object's position is described by a function of time, velocity is described by the time derivative of the position, , and the object's acceleration is described by the time derivative of the velocity, . and :
It is often convenient to discuss the average of the latter two quantities between times
and . Part A You throw a ball. Air resistance on the ball is negligible. Which of the following functions change with time as the ball flies through the air? Hint A.1 Newton's 2nd Law Hint not displayed ANSWER: only the position of the ball only the velocity of the ball only the acceleration of the ball the position and velocity of the ball the position and the velocity and acceleration of the ball
Part B You are driving a car at 65 mph. You are traveling north along a straight highway. What could you do to give the car a nonzero acceleration? Hint B.1 What constitutes a nonzero acceleration? Hint not displayed ANSWER: Press the brake pedal.
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Turn the steering wheel. Either press the gas or the brake pedal. Either press the gas or the brake pedal or turn the steering wheel. Part C A ball is lodged in a hole in the floor near the outside edge of a merry-go-round that is turning at constant speed. Which kinematic variable or variables change with time, assuming that the position is measured from an origin at the center of the merry-go-round? Hint C.1 Change of a vector Hint not displayed ANSWER: the position of the ball only the velocity of the ball only the acceleration of the ball only both the position and velocity of the ball the position and velocity and acceleration of the ball
Part D For the merry-go-round problem, do the magnitudes of the position, velocity, and acceleration vectors change with time? Hint D.1 Change of magnitude of a vector Hint not displayed ANSWER: yes no
Understanding acceleration in two directional motion
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
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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
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
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 to the right straight down downward to the right
Direction of Acceleration of Pendulum
Learning Goal: To understand that the direction of acceleration is in the direction of the change of the velocity, which is unrelated to the direction of the velocity. The pendulum shown makes a full swing from to . Ignore friction and assume that the string is massless. The eight labeled arrows represent directions to be referred to when answering the following questions.
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Part A Which of the following is a true statement about the acceleration of the pendulum bob, . ANSWER: is equal to the acceleration due to gravity. is equal to the instantaneous rate of change in velocity. is perpendicular to the bob's trajectory. is tangent to the bob's trajectory.
Part B What is the direction of when the pendulum is at position 1? Part B.1 Velocity at position 1
What is the velocity of the bob when it is exactly at position 1? ANSWER: Part B.2 =0 Velocity of bob after it has descended
What is the velocity of the bob just after it has descended from position 1? ANSWER: very small and having a direction best approximated by arrow D very small and having a direction best approximated by arrow A very small and having a direction best approximated by arrow H The velocity cannot be determined without more information.
Enter the letter of the arrow parallel to . ANSWER: H Part C What is the direction of at the moment the pendulum passes position 2? Hint C.1 Instantaneous motion
At position 2, the instantaneous motion of the pendulum can be approximated as uniform circular motion. What is the direction of acceleration for an object executing uniform circular motion? Enter the letter of the arrow that best approximates the direction of . ANSWER: C We know that for the object to be traveling in a circle, some component of its acceleration must be pointing radially inward. Part D What is the direction of when the pendulum reaches position 3? Part D.1 Velocity just before position 3 Part not displayed Part D.2 Velocity of bob at position 3 Part not displayed Give the letter of the arrow that best approximates the direction of . ANSWER: F Part E
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As the pendulum approaches or recedes from which position(s) is the acceleration vector almost parallel to the velocity vector . ANSWER: position 2 only positions 1 and 2 positions 2 and 3 positions 1 and 3
Horizontally moving object goes into free fall
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 at an altitude . The package is ejected horizontally in the negative x direction with speed relative to the plane. Assume is less than . The positive x and y directions are defined in the figure.
Part A Find the initial velocity of the package, , with respect to the ground. Part A.1 Find the initial velocity of the package in the plane's frame of reference
Find the initial velocity of the package in the plane's frame of reference, . Express your answer in terms of given variables, using as the unit vector in the x direction. ANSWER: Hint A.2 = Velocity of the plane with respect to the ground .
The frame of reference of the plane is moving with velocity
Express the initial velocity of the package in terms of given quantities, , , , and the magnitude of the acceleration due to gravity , using and for the unit vectors in the x and y directions. ANSWER: Part B How long will it take the package to reach sea level from the time it is ejected? Hint B.1 Initial velocity in the y direction Because the package is ejected horizontally, the vertical component of its initial velocity is zero. Hint B.2 Equation of motion in the y direction . =
Recall that the equation of motion in the y direction is
Express your answer in terms of given quantities. Neglect air resistance. ANSWER: =
Part C
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If the package were to land right on the island, at what horizontal distance released? Part C.1 Find the horizontal velocity
from the plane to the island should the package be
You know that the package is in the air for a time . What is the horizontal component of the package's velocity, , while the package is in the air? Express your answer in terms of ANSWER: = and .
Express the distance in terms of , , , and . ANSWER: =
Part D What is the horizontal distance Part D.1 from the plane to the island when the package hits the ground?
Find the distance the plane travels in time
The package is in the air for a time before it hits the island. How far does the plane travel in this time? Express the distance ANSWER: = Putting it all together in terms of , , and .
Hint D.2
In Part C you have found how far the plane is from the island when the package is dropped. In the previous hint you have found how far the plane travels from the drop point while the package is in the air. The distance of the plane to the island is the difference between the distance the plane travels and the distance the plane was away from the island at the moment of the drop. Express the distance in terms of , , and ANSWER: = but not in terms of or .
Part E Find the velocity vector of the package when it hits the ground. Express your answer in terms of , , , and , using and as the unit vectors in the x and y directions. ANSWER: =
Part F What is the speed Hint F.1 of the package when it hits the ground?
Definition of the magnitude of a vector is . Use this definition to find the speed, which is the magnitude of the velocity.
The magnitude of
Answer in terms of , , , and . ANSWER: =
Part G The damage to the package decreases with decreased impact speed. What ejection speed package?
minimizes damage the done to the
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ANSWER:
=
Of cource one can never push a package away with the speed as high as the speed of an airplane. So think of the solution to this part as a limiting case. Consequently, the condition given in the problem description is safe.
Horizontal Cannon on a Cliff
A cannonball is fired horizontally from the top of a cliff. The cannon is at height ground level, and the ball is fired with initial horizontal speed . above
Part A Assume that the cannon is fired at cannonball at the time ? Part A.1 and that the cannonball hits the ground at time . What is the y position of the
Equation of motion in y direction
with initial y position of and initial y velocity . The particle has an Suppose that a particle starts at time acceleration in the positive y direction. What is the particle's position at some later time ? ANSWER: Part A.2 = y postition of cannonball 0
Given your answer to the previous question, choose the correct equation for the y position of the cannonball at any time before it hits the ground (or ). ANSWER: Part A.3 = Time at which the ball hits the ground , the time at which the cannonball hits the ground, in terms of and .
Use this equation to find ANSWER: =
Express the y position of the cannonball in terms of . The quantities ANSWER: =
and should not appear in your answer.
Part B Given that the projectile lands a distance
from the cliff, as shown, find the initial speed of the projectile, .
Part B.1 Velocity component in x direction Consider just the horizontal component of the motion. What is the speed of the projectile? Express your answer in terms of ANSWER: = and .
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Now, substitute in the expression for Part B.2
that you should have found while working out Part A.
If you are stuck, use as a last resort from the following list.
If you're still stuck, choose the correct answer for ANSWER:
Express the initial speed in terms of , , and . ANSWER: =
Part C What is the y position of the cannonball when it is a distance Part C.1 Velocity component in x direction and (or use the answer given to the first hint in Part B if you have already from the hill?
What is the speed of the projectile in the x direction? Express your answer in terms of answered it). ANSWER: = Use the initial velocity to get the time
Hint C.2
Use the answer from the previous hint to find the time when the x position is . You should be able to express this time in terms of . Then, use the general equation for the cannonball's y position as a function of time. This should allow you to solve for the y position when the x position is . Express the position of the cannonball in terms of ANSWER: = only.
Standard projectile problems
Projectile Motion Tutorial
Learning Goal: Understand how to apply the equations for 1-dimensional motion to the y and x directions separately in order to derive standard formulae for the range and height of a projectile. A projectile is fired from ground level at time , at an angle with respect to the horizontal. It has an initial speed . In this problem we are assuming that the ground is level.
Part A Find the time Hint A.1 it takes the projectile to reach its maximum height.
A basic property of projectile motion
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Part B Find , the time at which the projectile hits the ground. Part B.1 Two possible approaches
There are two good ways to find the total flight time : either by invoking the symmetry of this problem (limited to projectiles fired over level ground) or by finding a general expression for , the y position of the projectile, and setting this equal to the height at the end of the trajectory, . The second method is more general (i.e., it would work if the projectile landed on a hill of height H). What is the value of in this problem? Express in terms of quantities given in the introduction. =0 Some needed kinematics , taking .
ANSWER: Part B.2
Give an expression for the height as a function of time, Part B.2.a Equation of motion
Part not displayed Express your answer in terms of , , , and . ANSWER: Hint B.3 = Solving for . By plugging in you can determine . From Part B.i you know the value of , which
You now know the equation for is equal to .
Express the time in terms of , , and . ANSWER: =
Part C Find , the maximum height attained by the projectile. Hint C.1 Equation of motion
Keep in mind the equation of motion for that you have found in Part B.ii. If you can't find the equation of motion and have not done Part B.ii, please finish this part now. Part C.2 When is the projectile at the top of its trajectory? will the projectile reach the maximum height?
At which time
Answer in terms of , , and . ANSWER: Hint C.3 = Finding .
Remember that
Express the maximum height in terms of , , and . ANSWER: =
Part D
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Find the total distance Hint D.1
(often called the range) traveled in the x direction; in other words, find where the projectile lands.
When does the projectile hit the ground?
The projectile reaches the ground at time . Part D.2 Where is the projectile as a function of time?
Give an expression for the x position of the particle as a function of time. Hint D.2.a Acceleration in x direction Hint not displayed Answer in terms of , , and . ANSWER: Hint D.3 = Answer not displayed Finding the range Hint not displayed Part D.4 A list of possible answers Part not displayed Express the range in terms of , , and . ANSWER: =
The actual formula for
is less important than how it is obtained:
1. Consider the x and y motion separately. 2. Find the time of flight from the y-motion 3. Find the x-position at the end of the flight - this is the range. If you remember these steps, you can deal with many variants of the basic problem, such as: a cannon on a hill that fires horizontally (i.e. the second half of the trajectory), a projectile that lands on a hill, or a projectile that must hit a moving target.
Projectile Motion--Conceptual
A cannon is fired from the top of a cliff as shown in the figure. Ignore drag (air friction) for this question. Take of the cliff. as the height
Part A Which of the paths would the cannonball most likely follow if the cannon barrel is horizontal? Part A.1 Find the y position as a function of time Part not displayed Part A.2 Interpreting your equation
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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 of the 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
The acceleration of the cannonball after it is fired is the constant acceleration due to gravity.
NOTE: In Part C of the Battleship Shells problem, notice it specifies that BOTH shells are launched at an angle GREATER than 45 degrees.
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 same speed. 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? ANSWER: straight line parabola hyperbola The shape cannot be determined.
Part B 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.
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.
Part D
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Which shell is launched with a greater vertical velocity, ? ANSWER: A B Both shells are launched with the same vertical velocity.
Part E Which shell is launched with a greater horizontal velocity, ? ANSWER: A B Both shells are launched with the same horizontal velocity.
Part F Which shell reaches the greater maximum height? Part F.1 What determines maximum height? Part not displayed ANSWER: A B Both shells reach the same maximum height.
Part G Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)? Hint G.1 Consider the limiting case Hint not displayed ANSWER: A B Both shells have the same travel time.
Problem 3.1
A squirrel has x- and y-coordinates ( 1.10 , 3.20 ) at time Part A For this time interval, find the x-component of the average velocity. ANSWER: Part B For this time interval, find the y-component of the average velocity. ANSWER: Part C Find the magnitude of the average velocity. ANSWER: Part D 1.62 m/s -1.06 m/s 1.23 m/s and coordinates ( 4.90 , -0.100 ) at time = 3.10 .
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Find the direction of the average velocity. ANSWER: 41.0 below the x-axis
Problem 3.3
A web page designer creates an animation in which a dot on a computer screen has a position of 4.30 2.90 4.50 . Part A and . Give your answer as a pair of components separated by a comma. Find the average velocity of the dot between For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. ANSWER: Part B Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. ANSWER: Part C . Give your answer as a pair of components separated by a comma. For example, if you Find the instantaneous velocity at think the x component is 3 and the y component is 4, then you should enter 3,4. ANSWER: Part D Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. ANSWER: = 11.6,4.50 cm/s = 5.80,4.50 cm/s = 0,4.50 cm/s = 5.80,4.50 cm/s
Problem 3.6
A dog running in an open field has components of velocity = 2.40 and = -1.40 at time = 11.6 . For the time interval from = 11.6 to = 24.5 , the average acceleration of the dog has magnitude 0.320 and direction 28.5 measured from the toward the . At time = 24.5 , Part A what are the x- and y-components of the dog's velocity? ANSWER: Part B ANSWER: Part C = 0.570 m/s = 6.03 m/s
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What is the magnitude of the dog's velocity? ANSWER: Part D What is the direction of the dog's velocity (measured from the ANSWER: 5.40 toward the )? 6.05 m/s
Summary
11 of 11 items complete (61.39% avg. score) 67.53 of 110 points
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