Chapter 9

Projectiles

When you throw a baseball the path of the baseball is an example of projectile motion. In the absence of air friction, the horizontal part of the velocity will not change while the vertical part of the velocity will be accelerated.

59

Enough with the simple one dimensional stuff.  Now lets have some fun with two dimensional situations.  Remember that the horizontal motion and the  vertical motion can be  treated individually and independently.

1. You are walking  atop the white cliffs of Dover arm in arm with Ned. For no apparent reason you pick up a rock and hurl it horizontally  (off the cliff) at 18 m/s.  You estimate that the  rock has a mass of 1.37 kg.

a)  How far does the rock travel horizontally in 1.5 s?

b)  How far has the rock traveled vertically after 1.5 s.

c)  How far away from your hand is the rock after 1.5 s?

d)  What is the vertical component of the velocity of the rock after 1.5 s?

e)  What is the horizontal component of the velocity of the rock after 1.5 s?

f)  What is the velocity of the rock (speed and direction) after 1.5 s?

2. Jill is experiencing one of those horrible no good very bad days.  To remove some of her frustrations she  kicks a 0.4 kg can of  Lutefisk horizontally at 5 m/s off a cliff 15 m high.

a)  How long will it take for the Can to land?

b)  How far from the cliff will it land?

c)  What is the horizontal component of the can’s  velocity just before impact?

d)  What is the vertical component of the can’s velocity just before impact?

3.  Another boring day in the oval office.  Since taking over things seem to be going so well that you are considering making the job a part time position.  To pass the time you roll a marble off the executive disk and observe it hitting the floor 1.0  m from the desk.  If the desk top is 0.75  m above the floor, how fast was the marble traveling when it left the desk?

4.  A rusted piece of metal drops off of a bridge.  How fast is it traveling in m/s 2.6 seconds after its release?

5. Brunhilda Broom,  in a fit of rage over a Government class, throws a 12 oz. can of Armadillo Meat  directly up at 8.6 m/s.  Her government teacher seeing this display, decides that Brunhilda deserves an "A" (after all throwing things at a problem is the big

60

government way).  Oh but we are doing physics now sooooo the questions are:

a)  How high does the can go?

b)  How long is the can in the air?

6. Your time travel device falters again and you find yourself in the middle of a field in Pennsylvania just before the battle of Gettysburg.  You are handed a Sharps repeating carbine and told to practice.  You aim the gun so that the barrel is horizontal with the ground and fire the 0.04 kg (650 grains) bullet with a muzzle velocity of 400 m/s (1,300 fps).  The barrel was 1.8 m above the horizontal when you fired.

a)  What will be the time of flight for the bullet?

b)  How far from the gun will the bullet hit the ground?

7. You have been selected to play the first softball game on the moon.  In preparation for the game you look in the back of your physics book and determine that objects near the moon's surface accelerate towards the moons surface with an acceleration of 1.6

m/s2. Your trip to the moon in uneventful.  You debark from the

shuttle and begin to warm up.You throw a softball straight up and time its flight.  It is in the air for 4.2 s.

a) What was the softball's initial velocity as it left your hand?

b)  How high did the softball go?

8. You start to climb to the top of a light tower (still on the moon's softball field) and as you approach the top a small moon rock (a memento perhaps) falls out of your uniform pocket.  With nothing better to do (Ned did not make the trip), your lab partner Brunhilda measures the velocity of the rock just before it hits the moon's surface.  Her data indicates that the rock had a speed of 57.4 m/s just before it hit the moon.

a)  How high up the tower were you when the rock fell out of your pocket?

b)  What is the mass of the rock?

9. Ok back to some fun stuff

Alkina (aboriginal name for the moon) fires a dart from her blowgun at 40 m/s.  She aims the gun at 30° above the horizontal.

a)   What is the dart's initial vertical velocity?

b)  What is the dart's initial horizontal velocity?

c)  How long is the dart in the air?

d)  How far down range does the dart land?  (ignore for the moment the distance from the blowgun to the ground)

61

10. It is the bottom of the 11th inning.  The score is still tied.  It is hero time.  The pitch, the swing, the clink of a ball hitting the aluminum.  The ball leaves your bat with a speed of 46 m/s (about 100 miles per hour) at an angle of 57° above the horizontal.  Ignoring air resistance and the distance the ball is above the ground when you smack it.

a)  How long is the ball in the air?

b)  What is the maximum height of the ball?

c)  Will it hit the outfield wall on the fly, drop short of the wall or go over the wall?  The wall is 102 m away and has a height of 2 m?

11. Study hall is boring.  Time to time travel again.  Poof, you land in feudal Scotland, a time for castles and gore.  You are inside the walls of a makeshift fortification when a clan leader, Mel, notices you.  "Nice kilt" he comments as he notices the plaid of your skirt.  He is trying hard to remember what clan is represented by that particular pattern.  You notice that he is also wearing a kilt.  Your begin to ponder one of the age old questions of all of Scotland.  Mel interrupts your pondering with a question.  "Can you make a catapult?".  To make a long story as short as Mel's kilt, you construct a catapult which can hurl a vat of burning pitch.  The mass of the pitch filled vat is almost 30 kg, and your preliminary estimate indicates that the pitch will be pitched at 18. m/s at an angle of  45° above the horizontal.

a)  How long will the burning pitch be in the air?

b)  What is the vertical component of the velocity of the burning pitch as it leaves your catapult?

c)  What is the horizontal component of the velocity of the burning pitch as it leaves your catapult?

d)  How far away from the catapult does the burning pitch land?

e)  What is the maximum height of the burning pitch during its flight?

f)  If the fortification walls are 3 m high, how far from the wall should you put the catapult to make sure that the burning pitch clears your own fort?

12. Linda's time travel machine leaves her near a sand pit  by Bruno NE.  No Mel, no kilts, no Scotland.  Linda has only her slingshot and an endless supply of gravel pebbles.  There are no other signs of life forms in the vicinity. With her trusty slingshot, Linda fires a pebble which is in the air for 5.0 seconds and lands 75 m away from where it took off.

a)  What was the initial vertical component of the pebble's velocity?

b)  What was the initial horizontal component of the pebble's velocity?

62

Two freshmen types are comparing their prowess in throwing a baseball.  Melinda Rose can throw a baseball with an initial speed of 20 m/s.  Her friend Nettles, can throw a baseball with a speed of 25 m/s.  They decide on a friendly contest to see who can throw a baseball the farthest distance.  If Melinda wins Nettles must take her on a romantic canoe trip in the Benson Park Lagoon (hey they are freshmen.... what do you expect).  If Nettles wins Melinda agrees to go to the stock car races with him on Sunday night. Nettles throws his baseball at an angle of 28° with respect to the horizontal while Melinda throws hers at an angle of 43°.

13.  Who wins? (and by how much)

14.  You pick up your Sharps rifle and fire a bullet into the air.  The velocity of the bullet is 300 m/s and you make a 3.5° angle with the horizontal with the barrel of the gun.

a)  How far does the bullet go down range before hitting the ground?

b)  If a  1.7 m tall person is standing 100 m away from you in the direction you were aiming, would you hit the person?

c)  What is the maximum height the bullet reaches?

d)  How long will the bullet be in the air?

15.  A 0.7 kg rock is thrown so that it flies for 4.3 seconds before hitting the lunar surface and it lands 36 m from its launching point.  What was the initial vertical and horizontal speed?

Look over what you are doing in the above solutions.  First you split each problem into two separate parts: a vertical one and a horizontal one.  In the case of projectile motion, the vertical velocity is the thing that gets the ball into the air.  The greater the vertical velocity the longer the ball will stay aloft.

The horizontal velocity has no effect whatsoever on how long the ball stays in the air.  It is the thing that determines how far forward the object will fly.  The range of the projectile is determined by a combination of the horizontal speed at which the projectile flies and the vertical speed which determines how long the object will be in the air.

16. You are in summer employment heaven.  The circus has hired you, fame, travel, excitement.  You report for your first day of work and discover that only two jobs are left.  The first is using a shovel to clean out the cages of the various animals, the second is to be the "Human Cannonball". You think about your options

for all of 3.2 x 10 -8 seconds and decide that a summer of being

shot out of a cannon is your cup of tea.  Looking at the setup you see that the canon is aimed at an angle of 42° with respect to

63

horizontal and the gage on the side indicates that your muzzle velocity will be 11.6 m/s.

a)  How far away from the canon should you place the safety net? (i.e. what will your range be?)

b)  How long will each flight last?

c)  What will be your maximum height off of the circus floor?

17. In 1922 one of the Zacchinis, a famous family of circus performers, was the first human cannonball to be shot across and arena and into a net.  Over time they increased the height and distance of the flight until in 1939 or 1940, Emanuel Zacchini soared over three Ferris wheels and through a horizontal distance of 225 ft  (69 m). There is some controversy over these numbers as  the  current world record for the farthest human cannonball flight is by David Smith Jr. on March 10, 2011, at Milan in Italy, previously held by his father. The father David "Cannonball" Smith Sr made the previous record of 200 ft 4 in (61.06 m),[2] on August 31, 2002, at The Steele County Free Fair, Owatonna, Minnesota in the United States. It is estimated Smith Sr. was travelling at over 70 miles per hour (110 km/h) during the flight. Anyway, assuming that the numbers for Emanuel’s flight are accurate:

a) Assuming that he was launched at an angle of 53° from the horizontal, what did his initial speed have to be in order to fly  69 m?

b)  The first Ferris wheel was about 15 m high and was placed 23 m in front of the cannon, by how much did Emanuel clear the top of the Ferris wheel during his flight?

c)  The third Ferris wheel was 46 m in front of the cannon, by how much did Emanuel clear this wheel?

d)  What was his time of flight?

The propulsion in the muzzle of the cannon for his flights was so severe that he underwent a momentary blackout.  If he landed during the blackout he risked landing in a odd position and breaking his neck.  Emanuel trained himself to wake quickly.

18.  Frustrated with finals, Penelope decides to release her tensions by bombarding the wall of the new gym with water balloons.  She fires or at 38° from the horizontal with a speed of 21.4 m/s.  If she stands 13.2 m away from the wall of the gym, how far up the building will the water balloon land?

Time to be a general.

Solve this for the asked for variable

64

19. A rock is kicked with velocity v off a cliff of height h.  The acceleration due to gravity is g.

a)  How long will the rock be in the air?

b)  How far from the base of the cliff (d)  will it land?

20.  A disgruntled auto worker pushes a small foreign import off a 22 m high cliff.  If the vehicle lands 13 m from the base of the cliff, how fast was it going when it left the cliff?

21. This summer Lisa's job is not as exciting as being the human cannon ball.  Chef at Chez Schad (at the base of Big Baldy), pays well but the hours are long and the enjoyment is minimal.  It is Tuesday and Lisa is bored.  She decides to crack walnuts by catapulting them into a wall a distance s away.  If her catapult launches the walnuts with a speed of v and an angle of Ø.

a)  What are the initial vertical and horizontal velocities?

b)  How long is the projectile in the air?

c)  How high up  (h)   do the walnuts strike the wall?

22. Megan is disgruntled.  Working at a fast food restaurant is not the glamorous, fun job she thought it would be.  She takes a day off and finds herself standing at the top of a sheer cliff.  She estimates the height of the cliff to be about 12 m.  Megan is 1.6 m tall.  She takes an Egg McMuffin out of her bag and hurls it at 18 m/s at an angle 26° above the horizontal.

a)  What are the Egg McMuffin's initial horizontal and vertical velocities?

b)  How long will it be in the air?

c)  How far from the cliff's edge  will it land?

23.  A projectile is fired vertically at 27 m/s.  How long will it be in flight if a bird catches it in mid air, while it is falling, 25 m from he ground?

24. You are standing on top of the White Cliffs of Dover (remember them) 77 m above the water below.

a)  You pick up a small rock and throw it vertically down at 12 m/s.  What will be the time of flight for the rock?

b)  You throw the next rock out into the channel.  You launch the rock with an initial velocity of 12 m/s at an angle of 28° above the horizontal.  What will be the time of flight of the rock?  How far from the base of the cliff will the rock land?

65

Finally, you decide to throw a rock straight up with a velocity of 19 m/s.  The rock barely clears the edge of the cliff as it comes down. and it lands with a resounding kersplash as it hits the water at the base of the cliff.

c)  How fast is the rock going when it hits the water?

d)  How long is the rock in the air?

25.  A bridge is a height h above the ground and the bridge is on a planet whose acceleration due to gravity is g.  How long (t) will it take a rock to fall if it is released from rest off the bridge?  How long (t) will it take if the rock is thrown up with a velocity vo?

39. An elephant is thrown at velocity vo at angle ø on level ground on a planet where the acceleration due to gravity is g.

a)  What is the elephant's initial vertical velocity (voy)?

b)  How long will it remain in the air  (t)?

c)  How far will it go before hitting the ground (d)?

d)  What is the elephants initial horizontal velocity (vox)?

e)  What is the maximum height the elephant will reach (h)?

26.  You are in your space shuttle, trying to catch up with Roberta  Baron, a vile thief who specializes in stealing hair scrunchies.

 Roberta  has a head start and is a distance "D" in front of you.  You know however that her space buggy can only travel at velocity v1.  You start from rest and can accelerate with an acceleration of a.  When will you catch Roberta Baron?

27.  Disgusted with Ned (oh the story you could tell.... save it for creative writing), you stand at the top of a tall cliff.  At your feet is a small rock whose shape reminds you of Ned's face.  You kick

the rock off of the cliff horizontally with a velocity of vo.  What the

the rock's speed be at time t ?

28.  This bowling thing is harder than it looks.  You launch your 5 kg bowling ball with a velocity of 22 m/s. Unfortunately you "loft" the ball at an angle of 65° above the horizontal.  How far down the alley will the ball land?

29.  How fast can you throw a baseball? (your results from the baseball toss lab).  At what angle should you throw a ball in order to have a maximum range (distance from you to where the ball lands)?  If you throw a ball at that angle how far will the ball travel before it hits the ground?

30.  You decide to have a water balloon fight in freshman hall (ok don't really do this).  The height of the hall is 3.2 m.  You can throw a water balloon and give it an initial velocity of about 15 m/s.  At what angle should you launch the balloon in order to maximize the range of the balloon (i.e. so that the balloon just misses the ceiling at the peak of its flight) ?  You know that your

66

friend Frieda can also throw a balloon at a speed of 15 m/s.  How far down the hall should you stand in order to be out of Frieda's range?

31. Jan, an inquisitive physics student, decides to investigate the properties of gravitational fields.  She drops a Pepsi machine from the top of the school.  She times the fall of the machine and determines that it takes 1.6 s to fall.

a)  How tall is the building?

She enlists a few of her friends and they push a candy machine across the roof of the building.  The candy machine acquires a speed of 2.4 m/s before it hurtles over the edge of the roof.

b)  How long will it take for the candy machine to hit the dirt?

c) With what horizontal component of velocity will it hit the dirt?

d)  With what vertical velocity does it hit?

e)  With what total velocity does it hit?

f)  At what angle does it smack into the dirt?

32. There you are holding a Twinkie in your hand, 30 m away from the dorm.  Your intended Twinkie target, Ned,  is standing by an open window.  You estimate the distance from the ground to the middle of the open window is about 8 m.  From your years in the

Twinkie Baseball League you know that you can still hurl a Twinkie with a speed of 40 m/s.

a)  At what angle should you launch your Twinkie in order to hit Nate?

b)  How fast will the Twinkie be going when it hits Ned?

33. Frannie  and Ned are cruising in her Lincoln Continental at a serene 31 m/s.  As they  pass by a convenience store,  a hooded terrorist jumps out and fires a heat seeking missile at the car.   

The missile is launched when the car is 200 m past the convenience store.  If the missile steadily accelerates at 17 m/s2, how many seconds will it take for the missile to reach the tail pipe of the Lincoln?

67

Fort Pulaski is located on Cockspur Island near Savannah Georgia. Construction was an engineering challenge as the weight of the 25,000,000 bricks in the fort had to be supported on a marshy ground. Pilings were buried 70 feet into the soft mud and arches constructed on top of the pilings.

Completed in 1847, it was basically unused until 1861 when the fort was occupied by Confederate forces.

In April of 1862, Union forces under Brigadier General Quincy Adams Gillmore established a battery of cannons on Tybee Island. Confederate forces were not concerned as the 2.5 mile distance from the Union cannons to the fort was more than traditional cannons could shoot with any type of accuracy.

Unknown to the Confederate forces, the yankees had installed rifled cannons called Parrot Rifles. By cutting a twisting groove inside the barrel of the cannons (an interesting engineering challenge on its own), the cannon projectiles would spin when in flight, giving the cannon projectile stability.

After 30 hours and more than 5,000 artillery rounds the walls of the fort were breached. One cannon shot just missed the 20 tons of powder stored at the fort. Considering his position to be hopeless, Confederate Colonel Charles Olmstead surrendered.

The rifled cannon established a new era of artillery warfare. The trajectories of cannon fire was now predictable.

68

Fort Pulaski is located on Cockspur Island near Savannah Georgia.