Chapter 03 Homework Assignment
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.
If an object's position is described by a function of time, . 0 (measured from a nonaccelerating reference frame), then the object's velocity is described by the time derivative of the position, 2 0 , and the object's acceleration is
described by the time derivative of the velocity, 0 2 0 . 0 .
0
It is often convenient to discuss the average of the latter two quantities between times 0 and 0 :
2BWH 0 0
and
BWH 0 0 .
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 1. The Pull of Gravity The reason the ball comes back to your hand is that it is being pulled on by the Earth's gravity. This is the same reason that the ball feel's heavy when it's resting in your hand. Does the weight of the ball change at different heights, or is the pull of gravity constant throughout the ball's flight? What does this tell you about the acceleration of the ball? ANSWER: Correct Part B 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
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 1. What constitutes a nonzero acceleration? The velocity of the car is described by a vector function, meaning it has both magnitude (65 mph) and direction (north). The car experiences a nonzero acceleration if you change either the magnitude of the velocity or the direction of the velocity. ANSWER: Correct Part C Press the brake pedal. 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.
A ball is lodged in a hole in the floor near the outside edge of a merrygoround 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 merrygoround?
Hint 1. Change of a vector A vector quantity has both magnitude and direction. The vector changes with time if either of these quantities changes with time. ANSWER: Correct Part D 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
For the merrygoround problem, do the magnitudes of the position, velocity, and acceleration vectors change with time?
Hint 1. Change of magnitude of a vector A vector quantity has both magnitude and direction. The magnitude of a vector changes with time only if the length changes with time. ANSWER: Correct yes no
PhET Tutorial: Projectile Motion
Learning Goal:
To understand how the trajectory of an object depends on its initial velocity, and to understand how air resistance affects the trajectory.
For this problem, use the PhET simulation Projectile Motion. This simulation allows you to fire an object from a cannon, see its trajectory, and measure its range and hang time (the amount of time in the air).
Start the simulation. Press Fire to launch an object. You can choose the object by clicking on one of the objects in the scrolldown menu at top right (a cannonball is not among the choices). To adjust the cannon barrel’s angle, click and drag on it or type in a numerical value (in degrees). You can also adjust the speed, mass, and diameter of the object by typing in values. Clicking Air Resistance displays settings for (1) the drag coefficient and (2) the altitude (which controls the air density). For this tutorial, we will use an altitude of zero (sea level) and let the drag coefficient be automatically set when the object is chosen.
Play around with the simulation. When you are done, click Erase and select a baseball prior to beginning Part A. Leave Air Resistance unchecked.
Part A
First, you will investigate purely vertical motion. The kinematics equation for vertical motion (ignoring air resistance) is given by
5 0 5 2 0ˆ #0 ,
where 5 is the initial position (which is 1.2 N above the ground due to the wheels of the cannon), 2 is the initial speed, and # is the acceleration due to gravity.
Shoot the baseball straight upward (at an angle of ¨) with an initial speed of 20 N T.
How long does it take for the baseball to hit the ground?
Express your answer with the appropriate units.
When the baseball is shot straight upward with an initial speed of 20 N T, what is the maximum height above its initial location? (Note that the ball’s initial height is denoted by the horizontal white line. It is initially 1.2 N above the ground. The yellow box that is below the target on the grass is measuring tape that should be used for this part.) Express your answer with appropriate units.
Hint 1. How to approach the problem
Use the measuring tape to determine the height. Align the plus sign at the beginning of the spool with the horizontal white line, and drag the end of the tape to the maximum height of the ball’s trajectory. You can zoom in or out using the magnification buttons above the Fire and Erase buttons.
ANSWER:
If the initial speed of the ball is doubled, how does the maximum height change?
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Erase all the trajectories, and fire the ball vertically again with an initial speed of 20 N T. As you found earlier, the maximum height is roughly 20 N. If the ball isn’t fired vertically, but at an angle less than ¨, it can reach the same maximum height if its initial speed is faster. Set the initial speed to 25 N T , and find the angle such that the maximum height is roughly 20 N. Experiment by firing the ball with many different angles. You can use the measuring tape to determine the maximum height of the trajectory and compare it to 20 N.
What is this angle?
In the previous part, you found that a ball fired with an initial speed of 25 N T and an angle of ¨ reaches the same height as a ball fired vertically with an initial speed of 20 N T. Which ball takes longer to land?
The figure shows two trajectories, made by two balls launched with different angles and possibly different initial speeds.
Based on the figure, for which trajectory was the ball in the air for the greatest amount of time? Hint 1. How to approach the problem Think about the result of Part E (think about the relationship between the maximum height of something thrown upward and the amount of time it is in the air). Does the time in the air depend on the range of the trajectory? ANSWER: Trajectory A It’s impossible to tell solely based on the figure. The balls are in the air for the same amount of time. Trajectory B
The range is the distance from the cannon when the ball hits the ground. This distance is given by the horizontal velocity (which is constant) times the amount of time the ball is in the air (which is determined by the vertical component of the initial velocity, as you just discovered).
Set the initial speed to 20 N T, and fire the ball several times while varying the angle between the cannon and the horizontal. Notice that the digital display near the top gives the range of the ball.
For which angle is the range a maximum (with the initial speed held constant)?
ANSWER:
How does the range of the object change if its initial velocity is doubled (keeping the angle fixed and less than ¨)?
ANSWER:
Part I
Now, let’s see what happens when the cannon is high above the ground. Click on the wheel of the cannon, and drag it upward as far as it goes (about 21 N above the ground). Set the initial velocity to 20 N T , and fire several balls while varying the angle.
For what angle is the range the greatest?
So far in this tutorial, you have been launching a baseball. Let’s see what happens to the trajectory if you launch something bigger and heavier, like a Buick car.
Compare the trajectory and range of the baseball to that of the Buick car, using the same initial speed and angle (e.g., ¨). (Be sure that air resistance is still turned off.) Which statement is true?
ANSWER:
Correct Since we are ignoring air resistance, the trajectory of the object does not depend on its mass or size. In the next part, you will turn on air resistance and discover what changes. Part K The trajectories and thus the range of the Buick and the baseball are identical. The trajectories differ; the range of the Buick is longer than that of the baseball. The trajectories differ; the range of the Buick is shorter than that of the baseball.
In the previous part, you discovered that the trajectory of an object does not depend on the object’s size or mass. But if you have ever seen a parachutist or a feather falling, you know this isn’t really true. That is because we have been neglecting air resistance, and we will now study its effects here.
Select Air Resistance for the simulation. Fire a baseball with an initial speed of roughly 20 N T and an angle of ¨. Compare the trajectory to the case without air resistance. How do the trajectories differ?
ANSWER:
Correct Air resistance is a force due to the object ramming through the air molecules, and is always in the opposite direction to the object’s velocity. This means the air resistance force will slow the object down, resulting in a shorter range (the simulation assumes the air is still; there is no strong tailwind). Part L The trajectories are identical. The trajectory with air resistance has a shorter range. The trajectory with air resistance has a longer range.
Notice that you can adjust the diameter (and mass) of any object (e.g., you can make a really big baseball). What happens to the trajectory (with air resistance on) when you increase the diameter while keeping the mass constant?
ANSWER:
Correct Since the surface area increases if the diameter increases, the object is sweeping through more air, causing more collisions, and a greater force of air drag (in fact, if the diameter is doubled, for a given speed, the force of air drag is increased by a factor of four). This greater force of air drag causes the object to slow down more quickly, resulting in a slower average speed and a shorter range. Part M Increasing the size makes the range of the trajectory decrease. The size of the object doesn’t affect the trajectory. Increasing the size makes the range of the trajectory increase.
You might think that it is never a good approximation to ignore air resistance. However, often it is. Fire the baseball without air resistance, and then fire it with air resistance (same angle and initial speed). Then, adjust the mass of the baseball (increase it and decrease it) and see what happens to the trajectory. Don’t change the diameter.
When does the range with air resistance approach the range without air resistance?
ANSWER:
The range with air resistance approaches the range without air resistance as the mass of the baseball is increased.
The range with air resistance approaches the range without air resistance as the mass of the baseball is decreased.
It never does. Regardless of the mass, the range with air resistance is always shorter than the range without.
Correct
As the mass is increased, the force of gravity on the baseball becomes larger. The force due to air drag just depends on the speed and the size of the object, so it doesn’t change if the mass changes. As the mass gets large enough, the force of gravity becomes much larger than the air drag force, and so the air drag force becomes negligible. This results in a trajectory nearly the same as when air resistance is turned off. Thus, for small, dense objects (like rocks and bowling balls), air resistance is typically unimportant, but for objects with a low density (like feathers) or a very large surface area (like parachutists), air resistance is very important.
PhET Interactive Simulations University of Colorado http://phet.colorado.edu
Video Tutor: Ball Fired from Cart on Incline
First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point.
Part A
Consider the video demonstration that you just watched. A more complete explanation of what you saw will be possible after covering Newton's laws. For now, consider the following question: How would the result of this experiment change if we replaced the ball with another one that had half the mass? Ignore air resistance.
Hint 1. How to approach this problem The kinematics of the situation in the video are actually quite complex; proving that the ball must land in the cart would not be easy. However, given that the ball did land in the cart in the video, you can answer the current question whether
the ball's mass affects the outcome quite easily. The ball is a projectile; the cart is an object moving in a straight line with constant acceleration. You know the kinematic equations that relate position, time, and velocity for this type of motion. If the ball's mass is a factor in a relevant equation, then it will affect the outcome; if not, then it won't. If you perform the experiment with a heavier ball, the ball's initial velocity will be lower, and the ball will therefore follow a shorter trajectory and strike the track closer to the trigger. But the kinematic equations tell you that the cart will still be at the same location as the ball when the ball reaches the track. ANSWER: Correct The ball lands in the cart regardless of its mass. The ball would land ahead of the cart. The ball would still land in the cart. The ball would land behind the cart.
Enhanced EOC: Exercise 3.7
The coordinates of a bird flying in the xy plane are given by 4 0 C0 and 5 0 NˆD0 , where C N T and D N T .
You may want to review ( pages 72 77) .
For help with math skills, you may want to review:
Differentiation of Polynomial Functions
Vector Magnitudes
Determining the Angle of a Vector
For general problemsolving tips and strategies for this topic, you may want to view a Video Tutor Solution of Calculating average and instantaneous accelerations.
Part A
Calculate the velocity vector of the bird as a function of time.
Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3t and the y component is 4t, then you should enter 3t,4t. Express your answer using two significant figures for all coefficients.
Hint 1. How to approach the problem
You are given the position functions of the bird in time, in both the x and y directions. How can you use those to find the velocities—as functions of time—for the bird in each direction, 24 0 and 25 0 ?
When you enter the x and y components of the velocity function, Mastering expects you’ll enter them
Calculate the acceleration vector of the bird as a function of time.
Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3t and the y component is 4t, then you should enter 3t,4t. Express your answer using two significant figures for all coefficients.
Calculate the magnitude of the bird's velocity at 0 ‘T.
Let the direction be the angle that the vector makes with the +x axis measured counterclockwise. Calculate the direction of the bird's velocity at 0 ‘T.
Express your answer in degrees using two significant figures.
Hint 1. How to approach the problem From Part A, you know the velocity components of the bird in x and y at any time . How can you find the direction of the resulting velocity vector from these two components? You might start by making a sketch of the velocity vectors in x and y (with the x axis horizontal), add them as vectors, and examine the angle of the resultant relative to the x axis. ANSWER: Correct Part E 0 = 300 J ¨
Calculate the magnitude of the bird's acceleration at 0 ‘T.
Express your answer using two significant figures.
‘T
Calculate the direction of the bird's acceleration at 0 ‘T.
Hint 1. How to approach the problem
From Part B, you know the acceleration components of the bird in x and y at any time 0. How can you find the direction of the resulting acceleration vector from these two components?
You might start by making a sketch of the acceleration vectors in x and y (with the x axis horizontal), add them as vectors, and examine the angle of the resultant relative to the x axis.
ANSWER:
Correct Part G = 270 J ¨
At 0 ‘T, is the bird speeding up, slowing down or moving at constant speed?
Hint 1. How to approach the problem Think about the direction of the velocity vector and the acceleration vector relative to each other. If they are the in the same direction, is the bird speeding up or slowing down? ANSWER: Correct speeding up slowing down moving at constant speed
Video Tutor: Ball Fired Upward from Accelerating Cart
First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point.
Part A
Consider the video you just watched. Suppose we replace the original launcher with one that fires the ball upward at twice the speed. We make no other changes. How far behind the cart will the ball land, compared to the distance in the original experiment?
Video Tutor: Balls Take High and Low Tracks
First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point.
Part A
Consider the video demonstration that you just watched. Which of the following changes could potentially allow the ball on the straight inclined (yellow) track to win? Ignore air resistance.
Select all that apply.
Hint 1. How to approach the problem
Answers A and B involve changing the steepness of part or all of the track. Answers C and D involve changing the mass of the balls. So, first you should decide which of those factors, if either, can change how fast the ball gets to the end of the track.
ANSWER:
Correct If the yellow track were tilted steeply enough, its ball could win. How might you go about calculating the necessary change in tilt? A. Increase the tilt of the yellow track. B. Make the downhill and uphill inclines on the red track less steep, while keeping the total distance traveled by the ball the same. C. Increase the mass of the ball on the yellow track. D. Decrease the mass of the ball on the red track.
Video Tutor: Range of a Gun at Two Firing Angles
First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point.
Part A
Which projectile spends more time in the air, the one fired from 30¨ or the one fired from 60¨ ?
Hint 1. How to approach this problem Which component of the initial velocity vector affects the time the projectile spends in the air? ANSWER: The one fired from 60 The one fired from 30 They both spend the same amount of time in the air. ¨ ¨
Correct The projectile fired from 60 has a greater vertical velocity than the one fired from 30 , so it spends more time in the air. ¨ ¨
Enhanced EOC: Exercise 3.13
A car comes to a bridge during a storm and finds the bridge washed out. The driver must get to the other side, so he decides to try leaping it with his car. The side the car is on is 22.5N above the river, whereas the opposite side is a mere 2.0N above the river. The river itself is a raging torrent 55.0N wide.
You may want to review ( pages 77 85) .
For help with math skills, you may want to review:
Vector Magnitudes
For general problemsolving tips and strategies for this topic, you may want to view a Video Tutor Solution of Different initial and final heights.
Part A
How fast should the car be traveling just as it leaves the cliff in order to just clear the river and land safely on the opposite side?
Hint 1. How to approach the problem
Start by drawing a diagram that shows the position of the car at takeoff and at landing on the other side of the river. Create an xy coordinate system that matches the diagram.
Consider that the car moves in projectile motion while it is in the air and that it initially moves (only) horizontally.
How long will it take for the car to fall vertically as it crosses the river? From this, how can you determine the velocity it must have had at takeoff to just make it across the river?
ANSWER:
What is the speed of the car just before it lands safely on the other side?
Hint 1. How to approach the problem How do the speeds in x and y change during the flight of the car across the river?
Enhanced EOC: Exercise 3.30
At its Ames Research Center, NASA uses its large “20G” centrifuge to test the effects of very large accelerations (hypergravity) on test pilots and astronauts. In this device, an arm 8.84 N long rotates about one end in a horizontal plane, and the astronaut is strapped in at the other end. Suppose that he is aligned along the arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this machine is typically 12.5 C . You may want to review ( pages 85 88) .
For general problemsolving tips and strategies for this topic, you may want to view a Video Tutor Solution of Centripetal acceleration on a carnival ride.
Part A
How fast must the astronaut's head be moving to experience this maximum acceleration?
Hint 1. How to approach the problem Consider that the astronaut is moving in uniform circular motion. How is the velocity related to the acceleration in this motion? ANSWER:
What is the difference between the acceleration of his head and feet if the astronaut is 2.00 N tall?
Hint 1. How to approach the problem In Part A, you used the acceleration experienced by the astronaut's head at a distance 8.84 from the N
How fast in SQN (revolutions per minute) is the arm turning to produce the maximum sustained acceleration?
Hint 1. How to approach the problem
From Part A, you know how fast the astronaut is moving linearly in N T (tangent to the revolving centrifuge). But what is the distance that the astronaut moves in each spin of the centrifuge?
Using these quantities, how can you determine the desired rate in revolutions per minute?
ANSWER:
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 ˆR to R . Ignore friction and assume that the string is massless.
The eight labeled arrows represent directions to be referred to when answering the following questions.
Part A
Which of the following is a true statement about the acceleration of the pendulum bob, .
ANSWER:
What is the direction of when the pendulum is at position 1?
Enter the letter of the arrow parallel to .
ANSWER: Correct Part C 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. H
What is the direction of at the moment the pendulum passes position 2?
Enter the letter of the arrow that best approximates the direction of .
Hint 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? ANSWER: Correct We know that for the object to be traveling in a circle, some component of its acceleration must be pointing radially inward. Part D C
What is the direction of when the pendulum reaches position 3?
Give the letter of the arrow that best approximates the direction of .
Hint 1. Velocity just before position 3 What is the velocity of the bob just before it reaches position 3? ANSWER:
Correct Part E F
As the pendulum approaches or recedes from which position(s) is the acceleration vector almost parallel to the velocity vector 2.
ANSWER: Correct position 2 only positions 1 and 2 positions 2 and 3 positions 1 and 3
± Arrow Hits Apple
An arrow is shot at an angle of J ¨ above the horizontal. The arrow hits a tree a horizontal distance N away, at the same height above the ground as it was shot. Use # N T for the magnitude of the acceleration due to gravity.
Part A
Find 0B , the time that the arrow spends in the air.
Answer numerically in seconds, to two significant figures.
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?
Express your answer numerically in seconds, to two significant figures.
Hint 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 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: ANSWER: = 1.1 0 G T
Correct = 5.6 0 E T
Enhanced EOC: Problem 3.59
According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy “Dizzy” Carlyle in a minor league game. The ball traveled 188 N (618 GU) before landing on the ground outside the ballpark.
You may want to review ( pages 77 85) .
For help with math skills, you may want to review:
Solving Quadratic Equations
Vector Magnitudes
Resolving Vector Components
For general problemsolving tips and strategies for this topic, you may want to view a Video Tutor Solution of A batted baseball.
Part A
Assuming the ball's initial velocity was 50¨ above the horizontal and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 N (3.0 GU) above ground level? Assume that the ground was perfectly flat.
Express your answer using two significant figures.
Hint 1. How to approach the problem
Start by drawing and labeling a diagram showing the starting location of the ball, the initial magnitude and direction of its motion, and where it lands. How far does it end up travelling horizontally? How far does it end up travelling vertically from its starting position? In what direction(s) does it accelerate?
Create an appropriate coordinate system to use for the problem, and identify the ball’s velocity components in terms of that coordinate system.
Consider that the ball moves in projectile motion once it leaves the bat and is in the air. Now, consider the five key kinematic variables for both the horizontal and vertical directions (displacement, initial velocity, final velocity, acceleration, and time). Which are given in the problem? Which aren’t known? How can you solve for the unknown quantities needed in terms of what is known?
How far would the ball be above a fence 3.0 N (10 GU) high if the fence was 116 N (380 GU) from home plate?
Express your answer using two significant figures.
Hint 1. How to approach the problem How long does it take the ball to travel to the fence? Use this to find the ball’s height at this time. ANSWER: Correct = 51 $ N
Problem 3.50
It is common to see birds of prey rising upward on thermals. The paths they take may be spirallike. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 6.00 N every 5.00 T and rises vertically at a rate of 3.00 N T.
Part A
Find the speed of the bird relative to the ground.
Express your answer using three significant figures.
Find the magnitude of the bird's acceleration.
Express your answer using three significant figures.
Find the direction of the bird's acceleration.
Express your answer using one significant figure.
Find the angle between the bird's velocity vector and the horizontal.
Express your answer using three significant figures.
ANSWER: Correct = 21.7 J 2 ¨
Conceptual Question 3.12
Part A
If you set the cruise control of your car to a certain speed and take a turn, the speed of the car will remain the same. Is the car accelerating?
ANSWER:
Correct No Yes
Problem 3.01
Part A
Object A has a position as a function of time given by ." 0 ‘ ‘ ‘N T 0‘?%‘ ‘ ‘N T 0 ‘?&. Object B has a position as a function of time given by .# 0 ‘ ‘ ‘N T 0‘?%‘ ‘ ˆ ‘N T 0 ‘?&. All quantities are SI units. What is the distance between object A and object B at time 0 = 2.00 T?
ANSWER:
Correct 11.5 m 8.25 m 4.95 m 9.90 m 6.60 m
Problem 3.05
Part A
An electron moves with a constant horizontal velocity of 3.0 × 106 m/s and no initial vertical velocity as it enters a deflector inside a TV tube. The electron strikes the screen after traveling 53 cm horizontally and 30 cm vertically upward with no horizontal acceleration. What is the constant vertical acceleration provided by the deflector? (The effects of gravity can be ignored.)
ANSWER:
Correct 1.9 × 10 13 m/s 2 3.4 × 10 3 m/s 2 6.4 × 10 1 m/s 2 9.6 × 10 12 m/s 2
Problem 3.16
Part A
A child throws a ball with an initial speed of 8.00 m/s at an angle of 40.0° above the horizontal. The ball leaves her hand 1.00 m above the ground and experience negligible air resistance.
(a) What is the magnitude of the ball's velocity just before it hits the ground?
ANSWER:
Correct Part B 9.14 m/s
(b) At what angle below the horizontal does the ball approach the ground?
ANSWER:
Correct Score Summary: Your score on this assignment is 91.1%. 47.9 °
You received 11.84 out of a possible total of 13 points.
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