Projectiles Day 1

Projectiles Day 1

Introduction to 2-Dimensional Motion

Introduction to 2D Motion

2-Dimensional Motion Definition: motion that occurs with

both x and y components. Example:

Playing pool . Throwing a ball to another person.

Each dimension of the motion can obey different equations of motion.

Solving 2-D Problems Resolve all vectors into components

x-component Y-component

Work the problem as two one-dimensional problems. Each dimension can obey different

equations of motion. Re-combine the results for the two

components at the end of the problem.

Sample Problem You run in a straight line at a speed of 5.0 m/s in a

direction that is 40o south of west.a) How far west have you traveled in 2.5 minutes?b) How far south have you traveled in 2.5 minutes?

Sample Problem A roller coaster rolls down a 20o incline with an

acceleration of 5.0 m/s2.a) How far horizontally has the coaster traveled in 10

seconds?b) How far vertically has the coaster traveled in 10 seconds?

Sample Problem A particle passes through the origin with a speed of

6.2 m/s in the positive y direction. If the particle accelerates in the negative x direction at 4.4 m/s2

a) What are the x and y positions at 5.0 seconds?

Sample Problem A particle passes through the origin with a speed of

6.2 m/s in the positive y direction. If the particle accelerates in the negative x direction at 4.4 m/s2

b) What are the x and y components of velocity at this time?

Projectiles Day 2

Projectiles

Demo Monkey gun – Sprott Physics DVD

Projectile Motion Something is fired, thrown, shot, or

hurled near the earth’s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored.

1-Dimensional Projectile Definition: A projectile that moves in a

vertical direction only, subject to acceleration by gravity.

Examples: Drop something off a cliff. Throw something straight up and catch it.

You calculate vertical motion only. The motion has no horizontal

component.

2-Dimensional Projectile Definition: A projectile that moves both

horizontally and vertically, subject to acceleration by gravity in vertical direction.

Examples: Throw a softball to someone else. Fire a cannon horizontally off a cliff. Shoot a monkey with a blowgun.

You calculate vertical and horizontal motion.

Horizontal Component of Velocity Is constant Not accelerated Not influence by gravity Follows equation: x = Vo,xt

Horizontal Component of Velocity

Vertical Component of Velocity Undergoes accelerated motion Accelerated by gravity (9.8 m/s2

down) Vy = Vo,y - gt y = yo + Vo,yt - 1/2gt2

Vy2 = Vo,y

2 - 2g(y – yo)

Horizontal and Vertical

Horizontal and Vertical

Projectiles Day 3

Projectiles with zero launch angle

(a.k.a.: Fired Horizontally)

Launch angle Definition: The angle at which a

projectile is launched. The launch angle determines what

the trajectory of the projectile will be. Launch angles can range from -90o

(throwing something straight down) to +90o (throwing something straight up) and everything in between.

Zero Launch angle

A zero launch angle implies a perfectly horizontal launch.

vo

Sample Problem The Zambezi River flows over Victoria Falls in Africa. The falls are

approximately 108 m high. If the river is flowing horizontally at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in freefall as it drops.

Sample Problem An astronaut on the planet Zircon tosses a rock horizontally

with a speed of 6.75 m/s. The rock falls a distance of 1.20 m and lands a horizontal distance of 8.95 m from the astronaut. What is the acceleration due to gravity on Zircon?

Sample Problem Playing shortstop, you throw a ball horizontally to the second

baseman with a speed of 22 m/s. The ball is caught by the second baseman 0.45 s later.

a) How far were you from the second baseman?b) What is the distance of the vertical drop?

Projectiles Day 4Horizontal Projectile Lab:

1. Roll marble down a ramp set up at the edge of a lab table…

2. WITHOUT rolling the marble off the table, calculate where it will hit the floor!

3. Mark your calculated landing spot on the floor with a target on cardboard

HINT: Calculate final velocity at the bottom of the ramp, use this velocity for the “firing” velocity when the marble is allowed to roll off the table. You also need change in Y.

Projectiles Day 5

Projectiles with General Launch Angle

General launch angle

vo

Projectile motion is more complicated when the launch angle is not straight up or down (90o or –90o), or perfectly horizontal (0o).

General launch angle

vo

You must begin problems like this by resolving the velocity vector into its components.

Resolving the velocity Use speed and the launch angle to find

horizontal and vertical velocity components

VoVo,y = Vo sin

Vo,x = Vo cos

Resolving the velocity Then proceed to work problems just like

you did with the zero launch angle problems.

VoVo,y = Vo sin

Vo,x = Vo cos

Sample problem A soccer ball is kicked with a speed of 9.50 m/s at an angle of

25o above the horizontal. If the ball lands at the same level from which is was kicked, how long was it in the air?

Sample problem Snowballs are thrown with a speed of 13 m/s from a roof

7.0 m above the ground. Snowball A is thrown straight downward; snowball B is thrown in a direction 25o above the horizontal. When the snowballs land, is the speed of A greater than, less than, or the same speed of B? Verify your answer by calculation of the landing speed of both snowballs.

Projectiles Day 6 Lab tomorrow

Projectiles launched over level ground These projectiles have highly

symmetric characteristics of motion. It is handy to know these

characteristics, since a knowledge of the symmetry can help in working problems and predicting the motion.

Lets take a look at projectiles launched over level ground.

Trajectory of a 2-D Projectile

x

y

Definition: The trajectory is the path traveled by any projectile. It is plotted on an x-y graph.

Trajectory of a 2-D Projectile

x

y

Mathematically, the path is defined by a parabola.

Trajectory of a 2-D Projectile

x

y

For a projectile launched over level ground, the symmetry is apparent.

Range of a 2-D Projectile

x

y

Range

Definition: The RANGE of the projectile is how far it travels horizontally.

Maximum height of a projectile

x

y

Range

MaximumHeight

The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.

Maximum height of a projectile

x

y

Range

MaximumHeight

The vertical velocity component is zero at maximum height.

Maximum height of a projectile

x

y

Range

MaximumHeight

For a projectile launched over level ground, the maximum height occurs halfway through the flight of the projectile.

Acceleration of a projectile

g

g

g

g

g

x

y

Acceleration points down at 9.8 m/s2 for the entire trajectory of all projectiles.

Velocity of a projectile

vo

vf

v

v

v

x

y

Velocity is tangent to the path for the entire trajectory.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

The velocity can be resolved into components all along its path.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

Notice how the vertical velocity changes while the horizontal velocity remains constant.

Velocity of a projectile

vy

vx

vx

vy

vx

vy

vx

x

y

vx

vy

Maximum speed is attained at the beginning, and again at the end, of the trajectory if the projectile is launched over level ground.

vo -

vo

Velocity of a projectile

Launch angle is symmetric with landing angle for a projectile launched over level ground.

to = 0

t

Time of flight for a projectile

The projectile spends half its time traveling upward…

Time of flight for a projectile

to = 0

t

2t

… and the other half traveling down.

Position graphs for 2-D projectiles

x

y

t

y

t

x

Velocity graphs for 2-D projectiles

t

Vy

t

Vx

Acceleration graphs for 2-D projectiles

t

ay

t

ax

““R2-D2 vs. The Swamp ThingR2-D2 vs. The Swamp Thing”” - - Luke Skywalker and R2-D2 are exploring Luke Skywalker and R2-D2 are exploring

the swamps of Dagobah when R2-D2 is the swamps of Dagobah when R2-D2 is eaten by a swamp creature… the creature eaten by a swamp creature… the creature doesndoesn’’t like R2t like R2’’s metallic taste, so it spits s metallic taste, so it spits him out. him out.

R2 comes flying out of the water with a R2 comes flying out of the water with a velocity of 22.5 m/s at an angle of +35 velocity of 22.5 m/s at an angle of +35 degrees with the ground! degrees with the ground!

How high does R2 get in the air, how long How high does R2 get in the air, how long is he in the air, and how far away does he is he in the air, and how far away does he land? land?

Luke is 6 ft (1.83 m) tall and standing 8 ft Luke is 6 ft (1.83 m) tall and standing 8 ft (2.44 m) from where R2 is spit out of the (2.44 m) from where R2 is spit out of the water. water.

If R2-D2 is spit in LukeIf R2-D2 is spit in Luke’’s direction, will he s direction, will he fly over Luke, or will they collide???fly over Luke, or will they collide???

If R2 misses Luke, find how close he was If R2 misses Luke, find how close he was over the top of Lukeover the top of Luke’’s head.s head.

If R2 flies into Luke, find how high he hits If R2 flies into Luke, find how high he hits him.him.

How fast is R2 going at LukeHow fast is R2 going at Luke’’s postion? s postion? (when he either hits Luke or flies (when he either hits Luke or flies overhead?)overhead?)

Where would Luke have to be standing Where would Luke have to be standing so that R2-D2 just barely flew over his so that R2-D2 just barely flew over his head?head?

Projectile Day 7

Projectile Lab

Projectile Day 8

More on projectile motion

The Range Equation Derivation is an important part of

physics. Your book has many more

equations than your formula sheet. The Range Equation is in your

textbook, but not on your formula sheet. You can use it if you can memorize it or derive it!

The Range Equation R = vo

2sin(2)/g. R: range of projectile fired over level

ground vo: initial velocity g: acceleration due to gravity : launch angle

Deriving the Range Equation

Sample problem A golfer tees off on level ground, giving the ball an

initial speed of 42.0 m/s and an initial direction of 35o above the horizontal.

a) How far from the golfer does the ball land?

Sample problem A golfer tees off on level ground, giving the ball an

initial speed of 42.0 m/s and an initial direction of 35o above the horizontal.

b) The next golfer hits a ball with the same initial speed, but at a greater angle than 45o. The ball travels the same horizontal distance. What was the initial direction of motion?