09 Projectiles - Apple
09 Projectiles - Apple
09 Projectiles - Apple
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Chapter 9<br />
<strong>Projectiles</strong><br />
When you throw a baseball the path of the baseball<br />
is an example of projectile motion. In the absence<br />
of air friction, the horizontal part of the velocity will<br />
not change while the vertical part of the velocity will<br />
be accelerated.
Enough with the simple one dimensional stuff. Now lets have<br />
some fun with two dimensional situations. Remember that the<br />
horizontal motion and the vertical motion can be treated<br />
individually and independently.<br />
1. You are walking atop the white cliffs of Dover arm in arm with<br />
Ned. For no apparent reason you pick up a rock and hurl it<br />
horizontally (off the cliff) at 18 m/s. You estimate that the rock<br />
has a mass of 1.37 kg.<br />
a) How far does the rock travel horizontally in 1.5 s?<br />
b) How far has the rock traveled vertically after 1.5 s.<br />
c) How far away from your hand is the rock after 1.5 s?<br />
d) What is the vertical component of the velocity of the rock after<br />
1.5 s?<br />
e) What is the horizontal component of the velocity of the rock<br />
after 1.5 s?<br />
f) What is the velocity of the rock (speed and direction) after 1.5<br />
s?<br />
2. Jill is experiencing one of those horrible no good very bad<br />
days. To remove some of her frustrations she kicks a 0.4 kg can<br />
of Lutefisk horizontally at 5 m/s off a cliff 15 m high.<br />
a) How long will it take for the Can to land?<br />
b) How far from the cliff will it land?<br />
c) What is the horizontal component of the can’s velocity just<br />
before impact?<br />
d) What is the vertical component of the can’s velocity just before<br />
impact?<br />
3. Another boring day in the oval office. Since taking over things<br />
seem to be going so well that you are considering making the job<br />
a part time position. To pass the time you roll a marble off the<br />
executive disk and observe it hitting the floor 1.0 m from the<br />
desk. If the desk top is 0.75 m above the floor, how fast was the<br />
marble traveling when it left the desk?<br />
4. A rusted piece of metal drops off of a bridge. How fast is it<br />
traveling in m/s 2.6 seconds after its release?<br />
5. Brunhilda Broom, in a fit of rage over a Government class,<br />
throws a 12 oz. can of Armadillo Meat directly up at 8.6 m/s. Her<br />
government teacher seeing this display, decides that Brunhilda<br />
deserves an "A" (after all throwing things at a problem is the big<br />
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government way). Oh but we are doing physics now sooooo the<br />
questions are:<br />
a) How high does the can go?<br />
b) How long is the can in the air?<br />
6. Your time travel device falters again and you find yourself in the<br />
middle of a field in Pennsylvania just before the battle of<br />
Gettysburg. You are handed a Sharps repeating carbine and told<br />
to practice. You aim the gun so that the barrel is horizontal with<br />
the ground and fire the 0.04 kg (650 grains) bullet with a muzzle<br />
velocity of 400 m/s (1,300 fps). The barrel was 1.8 m above the<br />
horizontal when you fired.<br />
a) What will be the time of flight for the bullet?<br />
b) How far from the gun will the bullet hit the ground?<br />
7. You have been selected to play the first softball game on the<br />
moon. In preparation for the game you look in the back of your<br />
physics book and determine that objects near the moon's surface<br />
accelerate towards the moons surface with an acceleration of 1.6<br />
m/s 2 . Your trip to the moon in uneventful. You debark from the<br />
shuttle and begin to warm up.You throw a softball straight up and<br />
time its flight. It is in the air for 4.2 s.<br />
a) What was the softball's initial velocity as it left your hand?<br />
b) How high did the softball go?<br />
8. You start to climb to the top of a light tower (still on the moon's<br />
softball field) and as you approach the top a small moon rock (a<br />
memento perhaps) falls out of your uniform pocket. With nothing<br />
better to do (Ned did not make the trip), your lab partner<br />
Brunhilda measures the velocity of the rock just before it hits the<br />
moon's surface. Her data indicates that the rock had a speed of<br />
57.4 m/s just before it hit the moon.<br />
a) How high up the tower were you when the rock fell out of your<br />
pocket?<br />
b) What is the mass of the rock?<br />
9. Ok back to some fun stuff<br />
Alkina (aboriginal name for the moon) fires a dart from her<br />
blowgun at 40 m/s. She aims the gun at 30° above the<br />
horizontal.<br />
a) What is the dart's initial vertical velocity?<br />
b) What is the dart's initial horizontal velocity?<br />
c) How long is the dart in the air?<br />
d) How far down range does the dart land? (ignore for the<br />
moment the distance from the blowgun to the ground)<br />
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10. It is the bottom of the 11th inning. The score is still tied. It is<br />
hero time. The pitch, the swing, the clink of a ball hitting the<br />
aluminum. The ball leaves your bat with a speed of 46 m/s (about<br />
100 miles per hour) at an angle of 57° above the horizontal.<br />
Ignoring air resistance and the distance the ball is above the<br />
ground when you smack it.<br />
a) How long is the ball in the air?<br />
b) What is the maximum height of the ball?<br />
c) Will it hit the outfield wall on the fly, drop short of the wall or go<br />
over the wall? The wall is 102 m away and has a height of 2 m?<br />
11. Study hall is boring. Time to time travel again. Poof, you land<br />
in feudal Scotland, a time for castles and gore. You are inside the<br />
walls of a makeshift fortification when a clan leader, Mel, notices<br />
you. "Nice kilt" he comments as he notices the plaid of your skirt.<br />
He is trying hard to remember what clan is represented by that<br />
particular pattern. You notice that he is also wearing a kilt. Your<br />
begin to ponder one of the age old questions of all of Scotland.<br />
Mel interrupts your pondering with a question. "Can you make a<br />
catapult?". To make a long story as short as Mel's kilt, you<br />
construct a catapult which can hurl a vat of burning pitch. The<br />
mass of the pitch filled vat is almost 30 kg, and your preliminary<br />
estimate indicates that the pitch will be pitched at 18. m/s at an<br />
angle of 45° above the horizontal.<br />
a) How long will the burning pitch be in the air?<br />
b) What is the vertical component of the velocity of the burning<br />
pitch as it leaves your catapult?<br />
c) What is the horizontal component of the velocity of the burning<br />
pitch as it leaves your catapult?<br />
d) How far away from the catapult does the burning pitch land?<br />
e) What is the maximum height of the burning pitch during its<br />
flight?<br />
f) If the fortification walls are 3 m high, how far from the wall<br />
should you put the catapult to make sure that the burning pitch<br />
clears your own fort?<br />
12. Linda's time travel machine leaves her near a sand pit by<br />
Bruno NE. No Mel, no kilts, no Scotland. Linda has only her<br />
slingshot and an endless supply of gravel pebbles. There are no<br />
other signs of life forms in the vicinity. With her trusty slingshot,<br />
Linda fires a pebble which is in the air for 5.0 seconds and lands<br />
75 m away from where it took off.<br />
a) What was the initial vertical component of the pebble's<br />
velocity?<br />
b) What was the initial horizontal component of the pebble's<br />
velocity?<br />
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Two freshmen types are comparing their prowess in throwing a<br />
baseball. Melinda Rose can throw a baseball with an initial speed<br />
of 20 m/s. Her friend Nettles, can throw a baseball with a speed<br />
of 25 m/s. They decide on a friendly contest to see who can<br />
throw a baseball the farthest distance. If Melinda wins Nettles<br />
must take her on a romantic canoe trip in the Benson Park<br />
Lagoon (hey they are freshmen.... what do you expect). If Nettles<br />
wins Melinda agrees to go to the stock car races with him on<br />
Sunday night. Nettles throws his baseball at an angle of 28° with<br />
respect to the horizontal while Melinda throws hers at an angle of<br />
43°.<br />
13. Who wins? (and by how much)<br />
14. You pick up your Sharps rifle and fire a bullet into the air. The<br />
velocity of the bullet is 300 m/s and you make a 3.5° angle with<br />
the horizontal with the barrel of the gun.<br />
a) How far does the bullet go down range before hitting the<br />
ground?<br />
b) If a 1.7 m tall person is standing 100 m away from you in the<br />
direction you were aiming, would you hit the person?<br />
c) What is the maximum height the bullet reaches?<br />
d) How long will the bullet be in the air?<br />
15. A 0.7 kg rock is thrown so that it flies for 4.3 seconds before<br />
hitting the lunar surface and it lands 36 m from its launching<br />
point. What was the initial vertical and horizontal speed?<br />
Look over what you are doing in the above solutions. First you<br />
split each problem into two separate parts: a vertical one and a<br />
horizontal one. In the case of projectile motion, the vertical<br />
velocity is the thing that gets the ball into the air. The greater the<br />
vertical velocity the longer the ball will stay aloft.<br />
The horizontal velocity has no effect whatsoever on how long the<br />
ball stays in the air. It is the thing that determines how far forward<br />
the object will fly. The range of the projectile is determined by a<br />
combination of the horizontal speed at which the projectile flies<br />
and the vertical speed which determines how long the object will<br />
be in the air.<br />
16. You are in summer employment heaven. The circus has hired<br />
you, fame, travel, excitement. You report for your first day of<br />
work and discover that only two jobs are left. The first is using a<br />
shovel to clean out the cages of the various animals, the second<br />
is to be the "Human Cannonball". You think about your options<br />
for all of 3.2 x 10 -8 seconds and decide that a summer of being<br />
shot out of a cannon is your cup of tea. Looking at the setup you<br />
see that the canon is aimed at an angle of 42° with respect to<br />
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horizontal and the gage on the side indicates that your muzzle<br />
velocity will be 11.6 m/s.<br />
a) How far away from the canon should you place the safety net?<br />
(i.e. what will your range be?)<br />
b) How long will each flight last?<br />
c) What will be your maximum height off of the circus floor?<br />
17. In 1922 one of the Zacchinis, a famous family of circus<br />
performers, was the first human cannonball to be shot across and<br />
arena and into a net. Over time they increased the height and<br />
distance of the flight until in 1939 or 1940, Emanuel Zacchini<br />
soared over three Ferris wheels and through a horizontal distance<br />
of 225 ft (69 m). There is some controversy over these numbers<br />
as the current world record for the farthest human cannonball<br />
flight is by David Smith Jr. on March 10, 2011, at Milan in Italy,<br />
previously held by his father. The father David "Cannonball" Smith<br />
Sr made the previous record of 200 ft 4 in (61.06 m),[2] on August<br />
31, 2002, at The Steele County Free Fair, Owatonna, Minnesota in<br />
the United States. It is estimated Smith Sr. was travelling at over<br />
70 miles per hour (110 km/h) during the flight. Anyway, assuming<br />
that the numbers for Emanuel’s flight are accurate:<br />
a) Assuming that he was launched at an angle of 53° from the<br />
horizontal, what did his initial speed have to be in order to fly 69<br />
m?<br />
b) The first Ferris wheel was about 15 m high and was placed 23<br />
m in front of the cannon, by how much did Emanuel clear the top<br />
of the Ferris wheel during his flight?<br />
c) The third Ferris wheel was 46 m in front of the cannon, by how<br />
much did Emanuel clear this wheel?<br />
d) What was his time of flight?<br />
The propulsion in the muzzle of the cannon for his flights was so<br />
severe that he underwent a momentary blackout. If he landed<br />
during the blackout he risked landing in a odd position and<br />
breaking his neck. Emanuel trained himself to wake quickly.<br />
18. Frustrated with finals, Penelope decides to release her<br />
tensions by bombarding the wall of the new gym with water<br />
balloons. She fires or at 38° from the horizontal with a speed of<br />
21.4 m/s. If she stands 13.2 m away from the wall of the gym,<br />
how far up the building will the water balloon land?<br />
Time to be a general.<br />
Solve this for the asked for variable<br />
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19. A rock is kicked with velocity v off a cliff of height h. The<br />
acceleration due to gravity is g.<br />
a) How long will the rock be in the air?<br />
b) How far from the base of the cliff (d) will it land?<br />
20. A disgruntled auto worker pushes a small foreign import off a<br />
22 m high cliff. If the vehicle lands 13 m from the base of the cliff,<br />
how fast was it going when it left the cliff?<br />
21. This summer Lisa's job is not as exciting as being the human<br />
cannon ball. Chef at Chez Schad (at the base of Big Baldy), pays<br />
well but the hours are long and the enjoyment is minimal. It is<br />
Tuesday and Lisa is bored. She decides to crack walnuts by<br />
catapulting them into a wall a distance s away. If her catapult<br />
launches the walnuts with a speed of v and an angle of Ø.<br />
a) What are the initial vertical and horizontal velocities?<br />
b) How long is the projectile in the air?<br />
c) How high up (h) do the walnuts strike the wall?<br />
22. Megan is disgruntled. Working at a fast food restaurant is not<br />
the glamorous, fun job she thought it would be. She takes a day<br />
off and finds herself standing at the top of a sheer cliff. She<br />
estimates the height of the cliff to be about 12 m. Megan is 1.6 m<br />
tall. She takes an Egg McMuffin out of her bag and hurls it at 18<br />
m/s at an angle 26° above the horizontal.<br />
a) What are the Egg McMuffin's initial horizontal and vertical<br />
velocities?<br />
b) How long will it be in the air?<br />
c) How far from the cliff's edge will it land?<br />
23. A projectile is fired vertically at 27 m/s. How long will it be in<br />
flight if a bird catches it in mid air, while it is falling, 25 m from he<br />
ground?<br />
24. You are standing on top of the White Cliffs of Dover<br />
(remember them) 77 m above the water below.<br />
a) You pick up a small rock and throw it vertically down at 12 m/<br />
s. What will be the time of flight for the rock?<br />
b) You throw the next rock out into the channel. You launch the<br />
rock with an initial velocity of 12 m/s at an angle of 28° above the<br />
horizontal. What will be the time of flight of the rock? How far<br />
from the base of the cliff will the rock land?<br />
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Finally, you decide to throw a rock straight up with a velocity of<br />
19 m/s. The rock barely clears the edge of the cliff as it comes<br />
down. and it lands with a resounding kersplash as it hits the water<br />
at the base of the cliff.<br />
c) How fast is the rock going when it hits the water?<br />
d) How long is the rock in the air?<br />
25. A bridge is a height h above the ground and the bridge is on<br />
a planet whose acceleration due to gravity is g. How long (t) will it<br />
take a rock to fall if it is released from rest off the bridge? How<br />
long (t) will it take if the rock is thrown up with a velocity vo?<br />
39. An elephant is thrown at velocity vo at angle ø on level ground<br />
on a planet where the acceleration due to gravity is g.<br />
a) What is the elephant's initial vertical velocity (voy)?<br />
b) How long will it remain in the air (t)?<br />
c) How far will it go before hitting the ground (d)?<br />
d) What is the elephants initial horizontal velocity (vox)?<br />
e) What is the maximum height the elephant will reach (h)?<br />
26. You are in your space shuttle, trying to catch up with Roberta<br />
Baron, a vile thief who specializes in stealing hair scrunchies.<br />
Roberta has a head start and is a distance "D" in front of you.<br />
You know however that her space buggy can only travel at<br />
velocity v1. You start from rest and can accelerate with an<br />
acceleration of a. When will you catch Roberta Baron?<br />
27. Disgusted with Ned (oh the story you could tell.... save it for<br />
creative writing), you stand at the top of a tall cliff. At your feet is<br />
a small rock whose shape reminds you of Ned's face. You kick<br />
the rock off of the cliff horizontally with a velocity of vo. What the<br />
the rock's speed be at time t ?<br />
28. This bowling thing is harder than it looks. You launch your 5<br />
kg bowling ball with a velocity of 22 m/s. Unfortunately you "loft"<br />
the ball at an angle of 65° above the horizontal. How far down<br />
the alley will the ball land?<br />
29. How fast can you throw a baseball? (your results from the<br />
baseball toss lab). At what angle should you throw a ball in order<br />
to have a maximum range (distance from you to where the ball<br />
lands)? If you throw a ball at that angle how far will the ball travel<br />
before it hits the ground?<br />
30. You decide to have a water balloon fight in freshman hall (ok<br />
don't really do this). The height of the hall is 3.2 m. You can<br />
throw a water balloon and give it an initial velocity of about 15 m/<br />
s. At what angle should you launch the balloon in order to<br />
maximize the range of the balloon (i.e. so that the balloon just<br />
misses the ceiling at the peak of its flight) ? You know that your<br />
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friend Frieda can also throw a balloon at a speed of 15 m/s. How<br />
far down the hall should you stand in order to be out of Frieda's<br />
range?<br />
31. Jan, an inquisitive physics student, decides to investigate the<br />
properties of gravitational fields. She drops a Pepsi machine from<br />
the top of the school. She times the fall of the machine and<br />
determines that it takes 1.6 s to fall.<br />
a) How tall is the building?<br />
She enlists a few of her friends and they push a candy machine<br />
across the roof of the building. The candy machine acquires a<br />
speed of 2.4 m/s before it hurtles over the edge of the roof.<br />
b) How long will it take for the candy machine to hit the dirt?<br />
c) With what horizontal component of velocity will it hit the dirt?<br />
d) With what vertical velocity does it hit?<br />
e) With what total velocity does it hit?<br />
f) At what angle does it smack into the dirt?<br />
32. There you are holding a Twinkie in your hand, 30 m away from<br />
the dorm. Your intended Twinkie target, Ned, is standing by an<br />
open window. You estimate the distance from the ground to the<br />
middle of the open window is about 8 m. From your years in the<br />
Twinkie Baseball League you know that you can still hurl a Twinkie<br />
with a speed of 40 m/s.<br />
a) At what angle should you launch your Twinkie in order to hit<br />
Nate?<br />
b) How fast will the Twinkie be going when it hits Ned?<br />
33. Frannie and Ned are cruising in her Lincoln Continental at a<br />
serene 31 m/s. As they pass by a convenience store, a hooded<br />
terrorist jumps out and fires a heat seeking missile at the car.<br />
The missile is launched when the car is 200 m past the<br />
convenience store. If the missile steadily accelerates at 17 m/s2,<br />
how many seconds will it take for the missile to reach the tail pipe<br />
of the Lincoln?<br />
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Fort Pulaski is located on Cockspur Island near Savannah<br />
Georgia. Construction was an engineering challenge as the<br />
weight of the 25,000,000 bricks in the fort had to be supported<br />
on a marshy ground. Pilings were buried 70 feet into the soft<br />
mud and arches constructed on top of the pilings.<br />
Completed in 1847, it was basically unused until 1861 when the<br />
fort was occupied by Confederate forces.<br />
In April of 1862, Union forces under Brigadier General Quincy<br />
Adams Gillmore established a battery of cannons on Tybee<br />
Island. Confederate forces were not concerned as the 2.5 mile<br />
distance from the Union cannons to the fort was more than<br />
traditional cannons could shoot with any type of accuracy.<br />
Unknown to the Confederate forces, the yankees had installed<br />
rifled cannons called Parrot Rifles. By cutting a twisting groove<br />
inside the barrel of the cannons (an interesting engineering<br />
challenge on its own), the cannon projectiles would spin when<br />
in flight, giving the cannon projectile stability.<br />
After 30 hours and more than 5,000 artillery rounds the walls of<br />
the fort were breached. One cannon shot just missed the 20<br />
tons of powder stored at the fort. Considering his position to<br />
be hopeless, Confederate Colonel Charles Olmstead<br />
surrendered.<br />
The rifled cannon established a new era of artillery warfare. The<br />
trajectories of cannon fire was now predictable.<br />
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Fort Pulaski is located on Cockspur Island near Savannah<br />
Georgia.<br />
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