## One-Dimensional, Constant Velocity

- You are writing a short adventure story for your
English class. In your story, two submarines on a
secret mission need to arrive at a place in the
middle of the Atlantic ocean at the same time.
They start out at the same time from positions
equally distant from the rendezvous point. They
travel at different velocities but both go in a
straight line. The first submarine travels at an
average velocity of 20 km/hr for the first 500
km, 40 km/hr for the next 500 km, 30 km/hr for
the next 500 km and 50 km/hr for the final 500
km. In the plot, the second submarine is required
to travel at a constant velocity, so the captain
needs to determine the magnitude of that
velocity.

- It is a beautiful weekend day and, since winter
will soon be here, you and four of your friends
decide to spend it outdoors. Two of your friends
just want to relax while the other two want some
exercise. You need some quiet time to study. To
satisfy everyone, the group decides to spend the
day on the river. Two people will put a canoe in
the river and just drift downstream with the 1.5
mile per hour current. The second pair will begin
at the same time as the first from 10 miles
downstream. They will paddle upstream until the
two canoes meet. Since you have been canoeing
with these people before, you know that they will
have an average velocity of 2.5 miles per hour
relative to the shore when they go against this
river current. When the two canoes meet, they
will come to shore and you should be there to
meet them with your van. You decide to go to that
spot ahead of time so you can study while you
wait for your friends. Where will you wait?

- It's a sunny Sunday afternoon, about 65 °F, and
you are walking around Lake Calhoun enjoying the
last of the autumn color. The sidewalk is crowded
with runners and walkers. You notice a runner
approaching you wearing a tee-shirt with writing
on it. You read the first two lines, but are
unable to read the third and final line before he
passes. You wonder, "Hmm, if he continues
around the lake, I bet I'll see him again, but I
should anticipate the time when we'll pass
again." You look at your watch and it is
3:07 p.m. You recall the lake is 3.4 miles in
circumference. You estimate your walking speed at
3 miles per hour and the runner's speed to be
about 7 miles per hour.

- You have joined the University team racing a
solar powered car. The optimal average speed for
the car depends on the amount of sun hitting its
solar panels. Your job is to determine strategy
by programming a computer to calculate the car's
average speed for a day consisting of different
race conditions. To do this you need to determine
the equation for the day's average speed based on
the car's average speed for each part of the
trip. As practice you imagine that the day's race
consists of some distance under bright sun, the
same distance with partly cloudy conditions, and
twice that distance under cloudy conditions.

- Because of your technical background, you have
been given a job as a student assistant in a
University research laboratory that has been
investigating possible accident avoidance systems
for oil tankers. Your group is concerned about
oil spills in the North Atlantic caused by a
super tanker running into an iceberg. The group
has been developing a new type of down-looking
radar which can detect large icebergs. They are
concerned about its rather short range of 2
miles. Your research director has told you that
the radar signal travels at the speed of light
which is 186,000 miles per second but once the
signal arrives back at the ship it takes the
computer 5 minutes to process the signal.
Unfortunately, the super tankers are such huge
ships that it takes a long time to turn them.
Your job is to determine how much time would be
available to turn the tanker to avoid a collision
once the tanker detects an iceberg. A typical
sailing speed for super tankers during the winter
on the North Atlantic is about 15 miles per hour.
Assume that the tanker is heading directly at an
iceberg that is drifting at 5 miles per hour in
the same direction that the tanker is going.

### The following four problems are mathematically equivalent, with different contexts.

- You and your friend run outdoors at least 10
miles every day no matter what the weather (well
almost). Today the temperature is at a brisk 0 oF
with a -20 oF wind chill. Your friend, a real
running fanatic, insists that it is OK to run.
You agree to this madness as long as you both
begin at your house and end the run at her nice
warm house in a way that neither of you has to
wait in the cold. You know that she runs at a
very consistent pace with an average speed of 3.0
m/s, while your average speed is a consistent 4.0
m/s. Your friend finishes warming up first so she
can get a head start. The plan is that she will
arrive at her house first so that she can unlock
the door before you arrive. Five minutes later,
you notice that she dropped her keys. If she
finishes her run first she will have to stand
around in the cold and will not be happy. How far
from your house will you be when you catch up to
her if you leave immediately, run at your usual
pace, and don't forget to take her keys?

- Because of your technical background, you have
been given a job as a student assistant in a
University research laboratory that has been
investigating possible accident avoidance systems
for oil tankers. Your group is concerned about
oil spills in the North Atlantic caused by a
super tanker running into an iceberg. The group
has been developing a new type of down-looking
radar which can detect large icebergs. They are
concerned about its rather short range of 2
miles. Your research director has told you that
the radar signal travels at the speed of light
which is 186,000 miles per second but once the
signal arrives back at the ship it takes the
computer 5 minutes to process the signal.
Unfortunately, the super tankers are such huge
ships that it takes a long time to turn them.
Your job is to determine how much time would be
available to turn the tanker to avoid a collision
once the tanker detects an iceberg. A typical
sailing speed for super tankers during the winter
on the North Atlantic is about 15 miles per hour.
Assume that the tanker is heading directly at an
iceberg that is drifting at 5 miles per hour in
the same direction that the tanker is going.

- Because of your technical background, you have
been given a job as a student assistant in a
University research laboratory that has been
investigating possible accident avoidance systems
for automobiles. You have just begun a study of
how bats avoid obstacles. In your study, a bat is
fitted with a transceiver that broadcasts the
bats velocity to your instruments. Your research
director has told you that the signal travels at
the speed of light which is 1.0 ft/nanosecond (1
nanosecond is 10-9 seconds). You know that the
bat detects obstacles by emitting a forward going
sound pulse (sonar) which travels at 1100 ft/s
through the air. The bat detects the obstacle
when the sound pulse reflect from the obstacle
and that reflected pulse is heard by the bat. You
are told to determine the maximum amount of time
that a bat has after it detects the existence of
an obstacle to change its flight path to avoid
the obstacle. In the experiment your instruments
tell you that a bat is flying straight toward a
wall at a constant velocity of 20.0 ft/s and
emits a sound pulse when it is 10.0 ft from the
wall.

- You have been hired to work in a University
research laboratory assisting in experiments to
determine the mechanism by which chemicals such
as aspirin relieve pain. Your task is to
calibrate your detection equipment using the
properties of a radioactive isotope (an atom with
an unstable nucleus) which will later be used to
track the chemical through the body. You have
been told that your isotope decays by first
emitting an electron and then, some time later,
it emits a photon which you know is a particle of
light. You set up your equipment to determine the
time between the electron emission and the photon
emission. Your apparatus detects both electrons
and photons. You determine that the electron and
photon from a decay arrive at your detector at
the same time when it is 2.0 feet from your
radioactive sample. A previous experiment has
shown that the electron from this decay travels
at one half the speed of light. You know that the
photon travels at the speed of light which is 1.0
foot per nanosecond. A nanosecond is 10-9
seconds.

## One Dimensional, Constant Acceleration

- You are part of a citizen's group evaluating the
safety of a high school athletic program. To help
judge the diving program you would like to know
how fast a diver hits the water in the most
complicated dive. The coach has his best diver
perform for your group. The diver, after jumping
from the high board, moves through the air with a
constant acceleration of 9.8 m/s2. Later in the
dive, she passes near a lower diving board which
is 3.0 m above the water. With your trusty stop
watch, you determine that it took 0.20 seconds to
enter the water from the time the diver passed
the lower board. How fast was she going when she
hit the water?

- As you are driving to school one day, you pass a
construction site for a new building and stop to
watch for a few minutes. A crane is lifting a
batch of bricks on a pallet to an upper floor of
the building. Suddenly a brick falls off the
rising pallet. You clock the time it takes for
the brick to hit the ground at 2.5 seconds. The
crane, fortunately, has height markings and you
see the brick fell off the pallet at a height of
22 meters above the ground. A falling brick can
be dangerous, and you wonder how fast the brick
was going when it hit the ground. Since you are
taking physics, you quickly calculate the answer.

- Because of your knowledge of physics, you have been
hired as a technical adviser on a new action movie. In one scene, the
hero pursues the villain up to the top of a bungee jump. The villain
creates a diversion by dropping a bottle filled with deadly gas. The
script calls for the hero to quickly strap on a 100 ft. bungee cord and
jump straight down to grab the bottle out of the air just as the bungee
cord begins to stretch. Your job is to determine the feasibility of the
stunt by finding the initial speed with which the hero length needs to
jump downward to catch the bottle. You estimate that the hero can react
to the villain's dropping the bottle by strapping on the bungee cord and
jumping in 2.0 seconds.

- You are helping a friend devise some challenging
tricks for the upcoming Twin Cities Freestyle
Skateboard Competition. To plan a series of
moves, he needs to know the rate that the
skateboard, with him on board, slows down as it
coasts up the competition ramp which is at 30º
to the horizontal. Assuming that this rate is
constant, you decide to have him conduct an
experiment. When he is traveling as fast as
possible on his competition skateboard, he stops
pushing and coasts up the competition ramp. You
measure that he typically goes about 95 feet in 6
seconds. Your friend weighs 170 lbs wearing all
of his safety gear and the skateboard weighs 6
lbs.

- You have a summer job working for a University
research group investigating the causes of the
ozone depletion in the atmosphere. The plan is to
collect data on the chemical composition of the
atmosphere as a function of the distance from the
ground using a mass spectrometer located in the
nose cone of a rocket fired vertically. To make
sure the delicate instruments survive the launch,
your task is to determine the acceleration of the
rocket before it uses up its fuel. The rocket is
launched straight up with a constant acceleration
until the fuel is gone 30 seconds later. To
collect enough data, the total flight time must
be 5.0 minutes before the rocket crashes into the
ground.

## One Dimensional, Constant Velocity and Constant Acceleration

- You have landed a summer job as the technical
assistant to the director of an adventure movie
shot here in Minnesota. The script calls for a
large package to be dropped onto the bed of a
fast moving pick-up truck from a helicopter that
is hovering above the road, out of view of the
camera. The helicopter is 235 feet above the
road, and the bed of the truck is 3 feet above
the road. The truck is traveling down the road at
40 miles/hour. You must determine when to cue the
assistant in the helicopter to drop the package
so it lands in the truck. The director is paying
$20,000 per hour for the chopper, so he wants you
to do this successfully in one take.

- Just for the fun of it, you and a friend decide
to enter the famous Tour de Minnesota bicycle
race from Rochester to Duluth and then to St.
Paul. You are riding along at a comfortable speed
of 20 mph when you see in your mirror that your
friend is going to pass you at what you estimate
to be a constant 30 mph. You will, of course,
take up the challenge and accelerate just as she
passes you until you pass her. If you accelerate
at a constant 0.25 miles per hour each second
until you pass her, how long will she be ahead of
you?

- In your new job, you are the technical advisor
for the writers of a gangster movie about Bonnie
and Clyde. In one scene Bonnie and Clyde try to
flee from one state to another. (If they got
across the state line, they could evade capture,
at least for a while until they became Federal
fugitives.) In the script, Bonnie is driving down
the highway at 108 km/hour, and passes a
concealed police car that is 1 kilometer from the
state line. The instant Bonnie and Clyde pass the
patrol car, the cop pulls onto the highway and
accelerates at a constant rate of 2 m/s2. The
writers want to know if they make it across the
state line before the pursuing cop catches up
with them.

- The University Skydiving Club has asked you to
plan a stunt for an air show. In this stunt, two
skydivers will step out of opposite sides of a
stationary hot air balloon 5,000 feet above the
ground. The second skydiver will leave the
balloon 20 seconds after the first skydiver but
you want them both to land on the ground at the
same time. The show is planned for a day with no
wind so assume that all motion is vertical. To
get a rough idea of the situation, assume that a
skydiver will fall with a constant acceleration
of 32 ft/sec2 before the parachute opens. As soon
as the parachute is opened, the skydiver falls
with a constant velocity of 10 ft/sec. If the
first skydiver waits 3 seconds after stepping out
of the balloon before opening her parachute, how
long must the second skydiver wait after leaving
the balloon before opening his parachute?

- Because parents are concerned that children are
learning "wrong" science from TV, you
have been asked to be a technical advisor for a
science fiction cartoon show on Saturday morning.
In the plot, a vicious criminal (Natasha Nogood)
escapes from a space station prison. The prison
is located between galaxies far away from any
stars. Natasha steals a small space ship and
blasts off to meet her partners somewhere in deep
space. The stolen ship accelerates in a straight
line at its maximum possible acceleration of 30
m/sec2. After 10 minutes all of the fuel is
burned up and the ship coasts at a constant
velocity. Meanwhile, the hero (Captain Starr)
learns of the escape while dining in the prison
with the warden's daughter (Virginia Lovely). Of
course he immediately (as soon as he finishes
dessert) rushes off the recapture Natasha. He
gives chase in an identical ship, which has an
identical maximum acceleration, going in an
identical direction. Unfortunately, Natasha has a
30 minute head start. Luckily, Natasha's ship did
not start with a full load of fuel. With his full
load of fuel, Captain Starr can maintain maximum
acceleration for 15 minutes. How long will it
take Captain Starr's ship to catch up to
Natasha's?

- Because parents are concerned that children are
learning "wrong" science from TV, you
have been asked to be a technical advisor for a
new science fiction show. The show takes place on
a space station at rest in deep space far away
from any stars. In the plot, a vicious criminal
(Alicia Badax) escapes from the space station
prison. Alicia steals a small space ship and
blasts off to meet her partners somewhere in deep
space. If she is to just barely escape, how long
do her partners have to transport her off her
ship before she is destroyed by a photon torpedo
from the space station? In the story, the stolen
ship accelerates in a straight line at its
maximum possible acceleration of 30 m/sec2. After
10 minutes (600 seconds) all of the fuel is
burned and the ship coasts at a constant
velocity. Meanwhile, the hero of this episode
(Major Starr) learns of the escape while dining
with the station's commander. Of course she
immediately rushes off to fire photon torpedoes
at Alicia. Once fired, a photon torpedo travels
at a constant velocity of 20,000 m/s. By that
time Alicia has a 30 minute (1800 seconds) head
start on the photon torpedo.

- You want to visit your friend in Seattle and decide to
take the train. Unfortunately, you are late getting to the train
station. You are running as fast as you can, but 30 meters ahead of you
the train begins to pull out. You can run at a maximum speed of 8 m/s
and the train is accelerating at 1 m/s/s. In 50 meters you will reach a
barrier. Can you catch up to your train?

- Because of your knowledge of physics, you have been assigned to investigate a train wreck between a fast moving passenger train and a slower moving freight train both going in the same direction. You have statements from the engineer of each train and the stationmaster as well as some measurements which you make. To check the consistency of each person's description of the events leading up to the collision, you decide to calculate the distance from the station that the collision should have occurred if everyone were telling what really happened and compare that with the actual position of the wreck which is 0.5 miles from the station. In this calculation you decide that you can ignore all reaction times. Here is what you know:

- The stationmaster claims that she noted that the freight train was behind schedule. As regulations require, she switched on a warning light just as the last car of the freight train passed her.
- The freight train engineer says he was going at a constant speed of 10 miles per hour.
- The passenger train engineer says she was going at the speed limit of 40 miles per hour when she approached the warning light. Just as she reached the warning light she saw it go on and immediately hit the brakes.
- The warning light is located so that a train gets to it 2.0 miles before it gets to the station.
- The passenger train slows down at a constant rate of 1.0 mile per hour for each minute as soon as you hit the brakes.

DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM AND DESCRIBE THE PHYSICS OF THE PROBLEM. DO NOT SOLVE THIS PROBLEM.

## Two Dimensional, Constant Acceleration (Projectile Motion)

- While on a vacation to Kenya, you visit the port
city of Mombassa on the Indian Ocean. On the
coast you find an old Portuguese fort probably
built in the 16th century. Large stone walls rise
vertically from the shore to protect the fort
from cannon fire from pirate ships. Walking
around on the ramparts, you find the fort's
cannons mounted such that they fire horizontally
out of holes near the top of the walls facing the
ocean. Leaning out of one of these gun holes, you
drop a rock which hits the ocean 3.0 seconds
later. You wonder how close a pirate ship would
have to sail to the fort to be in range of the
fort's cannon? Of course you realize that the
range depends on the velocity that the cannonball
leaves the cannon. That muzzle velocity depends,
in turn, on how much gunpowder was loaded into
the cannon. (a) Calculate the muzzle velocity
necessary to hit a pirate ship 300 meters from
the base of the fort. (b) To determine how the
muzzle velocity must change to hit ships at
different positions, make a graph of horizontal
distance traveled by the cannonball (range)
before it hits the ocean as a function of muzzle
velocity of the cannonball for this fort.

- Because of your knowledge of physics, you
have been hired as a consultant for a new
James Bond movie, "Oldfinger".
In one scene, Bond jumps horizontally off
the top of a cliff to escape a villain.
To make the stunt more dramatic, the
cliff has a horizontal ledge a distance h
beneath the top of the cliff which
extends a distance L from the vertical
face of the cliff. The stunt coordinator
wants you to determine the minimum
horizontal speed, in terms of L and h,
with which Bond must jump so that he
misses the ledge.

- Because of your knowledge of physics, you
have been hired as a consultant for a new
James Bond movie, "Oldfinger".
In one scene, Bond jumps horizontally off
the top of a cliff to escape a villain.
To make the stunt more dramatic, the
cliff has a horizontal ledge a distance h
beneath the top of the cliff which
extends a distance L from the vertical
face of the cliff. The stunt coordinator
wants you to determine the minimum
horizontal speed, in terms of L and h,
with which Bond must jump so that he
misses the ledge.
- You are on the target range preparing to shoot a
new rifle when it occurs to you that you would
like to know how fast the bullet leaves the gun
(the muzzle velocity). You bring the rifle up to
shoulder level and aim it horizontally at the
target center. Carefully you squeeze off the shot
at the target which is 300 feet away. When you
collect the target you find that your bullet hit
9.0 inches below where you aimed.

- You have a great summer job working on
the special effects team for a Minnesota
movie, the sequel to Fargo. A body is
discovered in a field during the fall
hunting season and the sheriff begins her
investigation. One suspect is a hunter
who was seen that morning shooting his
rifle horizontally in the same field. He
claims he was shooting at a deer and
missed. You are to design the
“flashback” scene which shows
his version of firing the rifle and the
bullet kicking up dirt where it hits the
ground. The sheriff later finds a bullet
in the ground. She tests the hunter's
rifle and finds the velocity that it
shoots a bullet (muzzle velocity). In
order to satisfy the nitpickers who
demand that movies be realistic, the
director has assigned you to calculate
the distance from the hunter that this
bullet should hit the ground as a
function of the bullet's muzzle velocity
and the rifle's height above the ground.

- You have a great summer job working on
the special effects team for a Minnesota
movie, the sequel to Fargo. A body is
discovered in a field during the fall
hunting season and the sheriff begins her
investigation. One suspect is a hunter
who was seen that morning shooting his
rifle horizontally in the same field. He
claims he was shooting at a deer and
missed. You are to design the
“flashback” scene which shows
his version of firing the rifle and the
bullet kicking up dirt where it hits the
ground. The sheriff later finds a bullet
in the ground. She tests the hunter's
rifle and finds the velocity that it
shoots a bullet (muzzle velocity). In
order to satisfy the nitpickers who
demand that movies be realistic, the
director has assigned you to calculate
the distance from the hunter that this
bullet should hit the ground as a
function of the bullet's muzzle velocity
and the rifle's height above the ground.
- The Police Department has hired you as a consultant in
a robbery investigation. A thief allegedly robbed a bank and, to escape
the pursing security guards, took the express elevator to the roof of
the building. Then, in order to not be caught with the evidence, the
thief allegedly threw the money bag to a waiting accomplice on the roof
of the next building. The defense attorney contends that in order to
reach the roof of that next building, the defendant would have had to
throw the money bag horizontally with a minimum velocity of 10
meters/second. However, in a test, the accused could throw the bag with
a maximum horizontal velocity of no more than 5 meters/second. How will
you advise the prosecuting attorney? You determine that the bank
building is 250 meters high, the next building is 100 meters high and
the distance between them is 20 meters.

- You are watching people practicing archery when
you wonder how fast an arrow is shot from a bow.
With a flash of insight you remember your physics
and see how you can easily determine what you
want to know by a simple measurement. You ask one
of the archers to pull back her bow string as far
as possible and shoot an arrow horizontally. The
arrow strikes the ground at an angle of 86
degrees from the vertical at 100 feet from the
archer.

- You read in the newspaper that rocks from Mars
have been found on Earth. Your friend says that the rocks were shot off
Mars by the large volcanoes there. You are skeptical so you decide to
calculate the magnitude of the velocity that volcanoes eject rocks from
geological evidence. You know the gravitational acceleration of objects
falling near the surface of Mars is only 40% that on the Earth. You can
look up the height of Martian volcanoes and determine the distance rocks
from a volcano hit the ground from pictures of the Martian surface. If
you assume the rocks farthest from a volcano were ejected at an angle of
45 degrees, what is the magnitude of the rock's velocity as a function
of its distance from the volcano and the height of the volcano for the
rock furthest from the volcano?

- You read in the newspaper that rocks from Mars
have been found on Earth. Your friend says that the rocks were shot off
Mars by the large volcanoes there. You are skeptical so you decide to
calculate the magnitude of the velocity that volcanoes eject rocks from
geological evidence. You know the gravitational acceleration of objects
falling near the surface of Mars is only 40% that on the Earth. You can
look up the height of Martian volcanoes and determine the distance rocks
from a volcano hit the ground from pictures of the Martian surface. If
you assume the rocks farthest from a volcano were ejected at an angle of
45 degrees, what is the magnitude of the rock's velocity as a function
of its distance from the volcano and the height of the volcano for the
rock furthest from the volcano?
- Watching the world series (only as an example of
physics in action), you wonder about the ability
of the catcher to throw out a base runner trying
to steal second. Suppose a catcher is crouched
down behind the plate when he observes the runner
breaking for second. After he gets the ball from
the pitcher, he throws as hard as necessary to
second base without standing up. If the catcher
throws the ball at an angle of 30 degrees from
the horizontal so that it is caught at second
base at about the same height as that catcher
threw it, how much time does it take for the ball
to travel the 120 feet from the catcher to second
base?

- Because of your physics background, you
have been hired as a consultant for a new
movie about Galileo. In one scene, he
climbs up to the top of a tower and, in
frustration over the people who ridicule
his theories, throws a rock at a group of
them standing on the ground. The rock
leaves his hand at 30º from the
horizontal. The script calls for the rock
to land 15 m from the base of the tower
near a group of his detractors. It is
important for the script that the rock
take precisely 3.0 seconds to hit the
ground so that there is time for a good
expressive close-up. The set coordinator
is concerned that the rock will hit the
ground with too much speed causing cement
chips from the plaza to injure one of the
high priced actors. You are told to
calculate that speed.

- Because of your physics background, you
have been hired as a consultant for a new
movie about Galileo. In one scene, he
climbs up to the top of a tower and, in
frustration over the people who ridicule
his theories, throws a rock at a group of
them standing on the ground. The rock
leaves his hand at 30º from the
horizontal. The script calls for the rock
to land 15 m from the base of the tower
near a group of his detractors. It is
important for the script that the rock
take precisely 3.0 seconds to hit the
ground so that there is time for a good
expressive close-up. The set coordinator
is concerned that the rock will hit the
ground with too much speed causing cement
chips from the plaza to injure one of the
high priced actors. You are told to
calculate that speed.
- While watching a softball game you see a play that
makes you wonder how fast a fielder can to react to a hit, run to the
fence, and leap up to make the catch. In this play, the batter hits a
ball when it is barely off the ground. It looks like a home run over the
left center field wall which is 200 ft from home plate. As soon as the
ball is hit, the left fielder runs to the wall, leaps high, and catches
the it just before it clears the top of 10 ft high wall. You estimate
that the ball left the bat at an angle of 30 degrees.

- You are still a member of a citizen's committee
investigating safety in the high school sports
program. Now you are interested in knee damage to
athletes participating in the long jump
(sometimes called the broad jump). The coach has
her best long jumper demonstrate the event for
you. He runs down the track and, at the take-off
point, jumps into the air at an angle of 30
degrees from the horizontal. He comes down in a
sand pit at the same level as the track 26 feet
away from his take-off point. With what velocity
(both magnitude and direction) did he hit the
ground?

- In your new job, you are helping to design stunts
for a new movie. In one scene the writers want a
car to jump across a chasm between two cliffs.
The car is driving along a horizontal road when
it goes over one cliff. Across the chasm, which
is 1000 feet deep, is another road at a lower
height. They want to know the minimum value of
the speed of the car so that it does not fall
into the chasm. They have not yet selected the
car so they want an expression for the speed of
the car, v, in terms of the car's mass, m, the
width of the chasm, w, and the height of the
upper road, h, above the lower road. The stunt
director will plug in the actual numbers after a
car is purchased.

- Your friend has decided to make some money during
the next State Fair by inventing a game of skill
that can be installed in the Midway. In the game
as she has developed it so far, the customer
shoots a rifle at a 5.0 cm diameter target
falling straight down. Anyone who hits the target
in the center wins a stuffed animal. Each shot
would cost 50 cents. The rifle would be mounted
on a pivot 1.0 meter above the ground so that it
can point in any direction at any angle. When
shooting, the customer stands 100 meters from
where the target would hit the ground if the
bullet misses. At the instant that the bullet
leaves the rifle (with a muzzle velocity of 1200
ft/sec according to the manual), the target is
released from its holder 7.0 meters above the
ground. Your friend asks you to try out the game
which she has set up on a farm outside of town.
Before you fire the gun you calculate where you
should aim.

- You have a summer job with an insurance company
and have been asked to help with the
investigation of a tragic "accident."
When you visit the scene, you see a road running
straight down a hill which has a slope of 10
degrees to the horizontal. At the bottom of the
hill, the road goes horizontally for a very short
distance becoming a parking lot overlooking a
cliff. The cliff has a vertical drop of 400 feet
to the horizontal ground below where a car is
wrecked 30 feet from the base of the cliff. Was
it possible that the driver fell asleep at the
wheel and simply drove over the cliff? After
looking pensive, your boss tells you to calculate
the speed of the car as it left the top of the
cliff. She reminds you to be careful to write
down all of your assumptions so she can evaluate
the applicability of the calculation to this
situation. Obviously, she suspects foul play.

- At your job with an insurance company, you have been
asked to help with the investigation of a tragic "accident." At the
scene is a road that runs straight down a hill with a slope of 10
degrees below the horizontal. At the bottom of the hill, the road goes
horizontally for a very short distance, then ends in a parking lot
overlooking a cliff. The cliff has a vertical drop of 400 feet to the
horizontal ground below where the wrecked car lies 30 feet from the base
of the cliff. The only witness claims that the car was parked somewhere
on the hill, he can't exactly remember where, and the car just began
coasting down the road. The witness did not hear an engine and thinks
that the driver was drunk and passed out knocking off his emergency
brake. The witness also remembers that the car took about 3 seconds to
get down the hill. The lead investigator drops a stone from the edge of
the cliff and, from the sound of it hitting the ground below, determines
that it takes 5.0 seconds to fall to the bottom. Based on that
information, you are told to calculate the car's average acceleration
coming down the hill using the statement of the witness and the other
facts in the case. You are reminded to write down all of your
assumptions so the investigation team can evaluate the applicability of
your calculation to this situation.

- Your group has been selected to serve on a
citizen's panel to evaluate a new proposal to
search for life on Mars. On this unmanned
mission, the lander will leave orbit around Mars
falling through the atmosphere until it reaches
10,000 meters above the surface of the planet. At
that time a parachute opens and takes the lander
down to 500 meters. Because of the possibility of
very strong winds near the surface, the parachute
detaches from the lander at 500 meters and the
lander falls freely through the thin Martian
atmosphere with a constant acceleration of 0.40g
for 1.0 second. Retrorockets then fire to bring
the lander to a softly to the surface of Mars. A
team of biologists has suggested that Martian
life might be very fragile and decompose quickly
in the heat from the lander. They suggest that
any search for life should begin at least 9
meters from the base of the lander. This biology
team has designed a probe which is shot from the
lander by a spring mechanism in the lander 2.0
meters above the surface of Mars. To return the
data, the probe cannot be more than 11 meters
from the bottom of the lander. Combining the data
acquisition requirements with the biological
requirements the team designed the probe to enter
the surface of Mars 10 meters from the base of
the lander. For the probe to function properly it
must impact the surface with a velocity of 8.0
m/s at an angle of 30 degrees from the vertical.
Can this probe work as designed?

- You have been hired as a technical
consultant for a new action movie. The
director wants a scene in which a car
goes up one side of an open drawbridge,
leaps over the gap between the two sides
of the bridge, and comes down safely on
the other side of the bridge. This
drawbridge opens in the middle by
increasing the angle that each side makes
with the horizontal by an equal amount.
The director wants the car to be stopped
at the bottom of one side of the bridge
and then accelerate up that side in an
amount of time which will allow for all
the necessary dramatic camera shots. He
wants you to determine the necessary
constant acceleration as a function of
that time, the gap between the two sides
of the open bridge, the angle that the
side of the open bridge makes with the
horizontal, and the mass of the car.

- You have been hired as a technical
consultant for a new action movie. The
director wants a scene in which a car
goes up one side of an open drawbridge,
leaps over the gap between the two sides
of the bridge, and comes down safely on
the other side of the bridge. This
drawbridge opens in the middle by
increasing the angle that each side makes
with the horizontal by an equal amount.
The director wants the car to be stopped
at the bottom of one side of the bridge
and then accelerate up that side in an
amount of time which will allow for all
the necessary dramatic camera shots. He
wants you to determine the necessary
constant acceleration as a function of
that time, the gap between the two sides
of the open bridge, the angle that the
side of the open bridge makes with the
horizontal, and the mass of the car.

## Two Dimensional, Constant Velocity and Constant Acceleration

### The following three problems have a very unfamiliar contexts.

- You are sitting in front of your TV waiting for
the World Series to begin when your mind wanders.
You know that the image on the screen is created
when electrons strike the screen which then gives
off light from that point. In the first TV sets,
the electron beam was moved around the screen to
make a picture by passing the electrons between
two parallel sheets of metal called electrodes.
Before the electrons entered the gap between the
electrodes, which deflect the beam vertically,
the electrons had a velocity of 1.0 x 106 m/s
directly toward the center of the gap and toward
the center of the screen. Each electrode was 5.0
cm long (direction the electron was going), 2.0
cm wide and the two were separated by 0.5 cm. A
voltage was applied to the electrodes which
caused the electrons passing between them to have
a constant acceleration directly toward one of
the electrodes and away from the other. After the
electrons left the gap between the electrodes
they were not accelerated and they continued
until they hit the screen. The screen was 15 cm
from the end of the electrodes. What vertical
electron acceleration between the electrodes
would be necessary to deflect the electron beam
20 cm from the center of the screen? DO ONLY THE
PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE
PROBLEM AND DESCRIBE THE PHYSICS OF THE PROBLEM.
DO NOT SOLVE THIS PROBLEM.

- You have a summer job in the cancer therapy division of
a hospital. This hospital treats cancer by hitting the tumor with high
energy protons from a cyclotron. When the protons leave the cyclotron
they are going at half the speed of light. You are in charge of
deflecting the protons so they hit the patent's tumor. This deflection
is accomplished by passing the proton beam between two flat, parallel
electrodes that have a length of 10 feet in the entering beam direction.
The protons enter the region between the electrodes going parallel to
their surface. The two electrodes are separated by 1.5 inches. A high
voltage is applied to the electrodes so that the protons passing between
them have a constant acceleration as they are attracted directly toward
one and repelled by the other. After the protons leave the region
between the plates, they are no longer accelerated during the remaining
200 feet to the patient. To set the correct high voltage, you need to
calculate the magnitude of the acceleration the protons need when they
are between the plates so that they are deflected by 1.0 degree, the
angle between the incident beam and the beam hitting the patient. The
speed of light is 1.0 foot per nanosecond.

- You have a summer job as an assistant in a
University research group that is designing a
devise to sample atmospheric pollution. In this
devise, it is useful to separate fast moving ions
from slow moving ones. To do this the ions are
brought into the device in a narrow beam so that
all of the ions are going in the same direction.
The ion beam then passes between two parallel
metal plates. Each plate is 5.0 cm long, 4.0 cm
wide and the two plates are separated by 3.0 cm.
A high voltage is applied to the plates causing
the ions passing between them to have a constant
acceleration directly toward one of the plates
and away from the other plate. Before the ions
enter the gap between the plates , they are going
directly toward the center of the gap parallel to
the surface of the plates. After the ions leave
the gap between the plates, they are no longer
accelerated during the 50 cm journey to the ion
detector. Your boss asks you to calculate the
magnitude of acceleration between the plates
necessary to separate ions with a velocity of 100
m/s from those in the beam going 1000 m/s by 2.0
cm?

- You are writing a short adventure story for your
English class. In your story, two submarines on a
secret mission need to arrive at a place in the
middle of the Atlantic ocean at the same time.
They start out at the same time from positions
equally distant from the rendezvous point. They
travel at different velocities but both go in a
straight line. The first submarine travels at an
average velocity of 20 km/hr for the first 500
km, 40 km/hr for the next 500 km, 30 km/hr for
the next 500 km and 50 km/hr for the final 500
km. In the plot, the second submarine is required
to travel at a constant velocity, so the captain
needs to determine the magnitude of that
velocity.