Chapter 3: Two Dimensional Motion

Section 1: Vectors

• scalar – a quantity with a magnitude, but no direction.– Speed is an example of a scalar.

• vector - a quantity with both a magnitude and direction.– Velocity & acceleration are vectors.

• Vectors are drawn with arrows on diagrams.

• vector component – one portion of a 2-dimensional vector.

• For example… Sarah is walking southeast at 1.7 m/s.– Her motion has a SOUTHWARD component, and an

EASTWARD component.

• SKETCH THIS:

• Vectors are resolved by separating them into their components.

South Component

East Component

“Resultant Vector”

Actual Motion

• Vectors may be added together…– resultant – a vector representing the sum of two or more

vectors.– When adding vectors, they must have the same unit.

• You cannot add 47.0 km west to 31 mi west.

Consider the two vectors shown below.When you add them together, you get…

9 cm + (-5 cm) = +4 cm OR -5 cm + 9 cm = +4 cm+4 cm is the resultant vector.

PositiveNegative ∆x2 = -5 cm

∆x = +9 cm

Bellringer 9/13

Which of the following quantities are scalars, and which are vectors?

a. The acceleration of a plane as it takes offb. Duration of flightc. Displacement of flightd. Number of passengers on the plane

• Vectors can be added graphically as well by drawing the vectors.

• Suppose Shannon walked 400 meters north to the store, and then 700 meters east to school. – To resolve vectors graphically, we use the head to tail

method.

∆y = 400m

∆x = 700m

d = 806mtail

head

1) ID the head & tail of the first vector.

2) Then slide the next vector’s tail to the first vector’s head.

3) Once all vectors are added, draw the resultant.

∆y = 400m

∆x = 700m

∆xresultant = 806m

Consider this scenario:A toy car is moving south at .80 m/s straight across a walkway that moves at 1.4 m/s towards the east. What will the car’s resultant speed be?

Draw this scenario. Use the scale 1cm = .1 m/s.Measure the resultant to find the answer.Try to find the answer using math…

• The car has a resultant speed of:• 1.61 m/s (toward the southeast)

• It is important to note that vectors can be moved parallel to themselves in a diagram.

• Moving vectors will be vital to drawing the correct resultant vector!

• Vectors can act at the same point at the same time.

• Coordinate System

y

-y

-x x

N

E

S

W

Please note that displacement may now be represented by “d”.

Vector Practice• A car is driven 125 km west, and then 65 km south.

Solve this problem both graphically & mathematically. Compare your answers.

(141 km)• Two shoppers exit the mall & walk 250.0 m down a

lane of cars, then turn 90 degrees and walk an additional 60.0 m. What is their displacement from the mall door?

(257 m)

Bellringer 9/13

• A plane can travel with a speed of 80mi/hr with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a

a. 10 mph headwindb. 10 mph tailwindc. 10 mph crosswindd. 60 mph crosswind

Practice

• Add the following vectors and determine the resultant. Use the graphical methodPart A: 10 km, North + 5 km, WestPart B: 30 km, West + 40 km, South

Review

• A roller coaster moves 85m horizontally, then travels 45 m at an angle of 30.0 degrees above the horizontal. What is its displacement from its starting point? (use graphical techniques)

Review

• A novice pilot sets a plane’s controls, thinking the plane will fly at 2.50x102 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity? Use graphical techniques.

204 km/hr, 75 degrees north of east

Practice

• Add the following vectors and determine the resultant:– 8.0m/s, 330° and 4.0 m/s, 45 °– 3.0m/s, 45 ° and 5.0m/s, 135 ° and 2.0m/s, 60 °– 2.0m/s, 315 ° and 5.0 m/s, 180 ° and 2.0m/s, 60 °

Chapter 3: Two Dimensional Motion

Section 2: Vector Operations

• Although finding vectors graphically is “fun”, it is not very practical.

• Vectors can also be determined using geometry

Pythagorean’s Theorem for Rt. Triangles:c2 = a2 + b2

c represents the hypotenuse; a & b represent legs

ca

b

The Pythagorean theorem will only give you the magnitude of the resultant vector!

• Pythagorean’s theorem can only be used when a right triangle is present!

• Direction is represented with angles.– Theta (θ) is used to represent the angle when working

with vectors.

– Trigonometry is applied in order to find the angles of right triangles. We’ll begin with tangent.

Tangent Function for Rt. Triangles:tan θ = opp/adj

(opp = opposite leg of angle; adj = leg adjacent to angle)

θ

hypopp

adj

• Vectors will often need to be broken down into components.– Definition: vector component – the projection of a

vector along the x & y axis.

The vector C can be broken down into it’s X component and Y component.

C

Sine Function for Rt. Triangles:sin θ = opp/hyp

(opp = opposite leg of angle; adj = leg adjacent to angle)

θ

hypopp

Cosine Function for Rt. Triangles:cos θ = adj/hyp

(opp = opposite leg of angle; adj = leg adjacent to angle)

Practice

• An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136m and its width is 230 m. What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top?

Resolving vectors into components

• X – component – parallel to x axis• Y – component – parallel to y axis• Example: A plane travels at 95 km/hr at an

angle of 20 degrees relative to the ground. Attempting to film the plane from below, a camera team travels in a truck that is directly beneath the plane. What velocity must the truck maintain to stay beneath the plane?

89.3 km/hr

Adding three or more right angle vectors

• A student drives his car 6.0k North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of displacement of the student?

Adding three or more right angle vectors

• Draw a diagram of the motion:

• Add together using head to tail:

Adding three or more right angle vectors

• Rearrange the order:

• Now, apply the Pythagorean theorem

10 km

Example

• Mac and Tosh are doing a Vector Walk Lab. Starting at the door of their physics classroom, they walk 2.0 meters, south. They make a right hand turn and walk 16.0 meters, west. They turn right again and walk 24.0 meters, north. They then turn left and walk 36.0 meters, west. What is the magnitude of their overall displacement?

56.5m

Bellringer 9/16

• While following the directions on a treasure map, a pirate walks 45.0m north and then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure? (include displacement and angle)

Adding vectors that are not perpendicular

• Cannot apply Pythagorean theorem or trig functions

• Determining magnitude and the direction of the resultant can be achieved by resolving each displacement vectors into its x and y components

• Components along each axis can be added together

Example

• A hiker walks 27.0 km from her base camp at 35 degrees south of east. The next day, she walks 41.0 km in a direction 65 degrees north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

Example

• A football player runs directly down the field for 35 m before turning to the right at an angle of 25 degrees from his original direction and running an additional 15m before getting tackled. What is the magnitude and direction of the runner’s total displacement?

Bellringer 2/24

• A plane travels 2.5 km at an angle of 35 degrees to the ground and then changes direction and travels 5.2km at an angle of 22 degrees to the ground. What is the magnitude and direction of the plane’s total displacement? (solve this algebraically)

Demo

• Try out these three motions and record your results:– Toss the tennis ball straight up while standing still– Toss the tennis ball then start walking forward at a

constant pace– Toss the tennis ball straight up while walking in a

straight line at a constant pace

Relative Motion

• Objects move within a medium that is moving with respect to an observer.

Relative MotionConsider a stunt dummy dropped from a plane.(a) When viewed from the plane, the stunt dummy falls straight down. (b) When viewed from a stationary position on the ground, the stunt dummy

follows a parabolic projectile path.

Example 1

• A motorboat traveling 4 m/s, East encounters a current traveling 3.0 m/s North.– What is the resultant velocity of the motorboat?– If the width of the river is 80 meters wide, then

how much time does it take the boat to travel shore to shore?

– What distance downstream does the boat reach the opposite shore?

Example 1 cont

Example 2

• A boat heading north crosses a wide river with a velocity of 10.00km/hr relative to the water. The river has a uniform velocity of 5.00 km/hr due East. Determine the boat’s velocity with respect to an observer on shore.

Example 3

A motorboat traveling 6 m/s, East encounters a current traveling 3.8 m/s, South.

a. What is the resultant velocity of the motor boat?

b. b. If the width of the river is 120 meters wide, then how much time does it take the boat to travel shore to shore?

• c. What distance downstream does the boat reach the opposite shore?

Bellringer 9/19

• During a rodeo, a clown runs 8.0 m north, turns 55 degrees north of east, and runs 3.5 m. Then, after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clown’s total displacement?

Chapter 3: Two Dimensional Motion

Section 3: Projectile Motion

vy

Vx

• Definition: projectile – any object launched into the air or dropped that accelerates vertically due to gravity.– Examples: A baseball, rocket, arrow, bullet, etc…

– Projectile motion describes the curved path of a projected object near Earth’s surface.

• Remember… free fall motion is motion in which gravity is the only vertical force acting on an object.– Free fall includes objects dropped AND objects thrown

upward.

Free Fall NonFree Fall

Free Fall

• Projectiles that have an initial horizontal velocity will always follow a parabolic path.

• Projectile motion is just freefall motion with a horizontal component.

The path of the projectile will always be a parabola.

• Even with an initial velocity, a projectile still falls at an acceleration of -9.81 m/s2

• Notice the path of the two projectiles.

• The equations of motion can be used to analyze projectiles that are dropped or launched horizontally:

Please note that these are not “new” equations. They are rearranged kinematic equations.

Vertical Motion of Projectiles with no initial vertical velocity.

Vy,f = ay∆t [ay=gravity, g]

Vy,f2 = 2g∆y

∆ y = ½g∆t2

• It is always assumed that the horizontal velocity of a projectile remains constant.

Horizontal Motion of a Projectile∆x = vx∆t

Note the horizontal

velocity (vx) of the object.

• Suppose a ball is pushed off the table, as shown.

• If the table is .75 m high, and the ball’s displacement is .58 m from the edge, how fast was it moving when it left the table?

Given:ay=g = -9.81 m/s2

∆y = .75m∆x = .58 mvx=?

1. Use ∆ y = ½g∆t2 and solve for time. (.39 s)2. Use ∆x = vx∆t and solve for vx.

Vx = 1.5 m/s

Example

• The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The magnitude of the rock’s horizontal displacement is 45.0m Find the speed at which the rock was kicked.

Bellringer 9/23

A pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0m horizontally before it hits the water below. What is the pelican’s speed?

• In order to assess the motion of a projectile launched at an angle, the horizontal & vertical components must be isolated.– Use trigonometry to break down the motion into

horizontal motion and vertical motion.

vy

vx

The projectile will continue in a parabolic path…But we can assess a portion of it’s path with trig.

• For a projectile launched at some angle, the initial horizontal velocity remains constant.

vx, i = vicosθ

v i

vy,i = visinθ

θ

• When you substitute the trigonometric values, you receive the equations for a projectile launched at some angle above the ground.

Vertical Motion of Projectiles Launched at an Anglevy,f = visinθ + ay∆t

vy,f2 = vi

2 (sinθ)2

+ 2g∆y

∆ y = (visinθ) ∆t + ½g∆t2

∆x = (vicosθ) ∆t

Example

• A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10.0m from the light pole, which is 5.00m high. The tip of her gun is 1.00m above the ground. At the same moment that the monkey drops a banana, the zookeeper shoots. If the dart travels at 50.0 m/s, will the dart hit the monkey, the banana, or neither one?

Example

• In a scene in an action movie, a stuntman jumps from the top of one building to the top of another building 4.0m away. After a running start, he leaps at a velocity of 5.0m/s at an angle of 15 degrees with respect to the flat roof. Will he make it to the other roof, which is 2.5m shorter than the building he jumps from?

Bellringer 9/24

• A baseball is thrown at an angle of 25 degrees relative to the ground at a speed of 23.0m/s. If the ball was caught 42.0m from the thrower, how long was it in the air? How high did the ball travel before being caught?

Review

Relative Motion

• Objects move within a medium that is moving with respect to an observer.

Example 1

• A motorboat traveling 4 m/s, East encounters a current traveling 3.0 m/s North.– What is the resultant velocity of the motorboat?– If the width of the river is 80 meters wide, then

how much time does it take the boat to travel shore to shore?

– What distance downstream does the boat reach the opposite shore?

Example 1 cont

Example 2

• A boat heading north crosses a wide river with a velocity of 10.00km/hr relative to the water. The river has a uniform velocity of 5.00 km/hr due East. Determine the boat’s velocity with respect to an observer on shore.

Example 3

A motorboat traveling 6 m/s, East encounters a current traveling 3.8 m/s, South.

a. What is the resultant velocity of the motor boat?

b. If the width of the river is 120 meters wide, then how much time does it take the boat to travel shore to shore?

• c. What distance downstream does the boat reach the opposite shore?

Bellringer 2/26

• While grazing, a caribou treks 125 m at 35.0° west of south, and then 105 m west. What is the caribou’s resultant displacement?

Review

• What is the difference between vector and scalar quantities?

Review

• A plane flies from city A to city B. City B is 1540 km west and 1160 km south of city A. What is the total displacement of the plane?

Review

• A dog walks 28 steps north and then walks 55 steps west to bury a bone. If the dog walks back to the starting point in a straight line, how many steps will the dog take? Use the graphical method to find the magnitude of the net displacement.

Review

• The figure above shows the path of a ball tossed from a building. Air resistance is ignored.

a. In the figure above, what would happen to the width of the ball’s path if it were launched with a greater velocity?

b. In the figure above, what would happen to the height of the ball’s path if it were launched with a greater velocity?

c. In the figure above, what would happen to the height of the ball’s path if it were launched with a lesser velocity?

Review

At what point of the ball’s path shown in the figure is the vertical component of the ball zero?

At what point is the magnitude of the ball’s velocity the least? The greatest?

Review

• A model rocket flies horizontally off the edge of a cliff at a velocity of 50.0 m/s. If the canyon below is 100.0 m deep, how far from the edge of the cliff does the model rocket land?

Review

• A baseball is thrown at an angle of 25 degrees to the ground at a speed of 23.0m/s. If the ball was caught 42 m from the thrower, how long was it in the air? How high did it travel before being caught?

Review

• A passenger on a bus moving east sees a man standing on a curb. From the passenger’s perspective, the man appears to

a. stand still.b. move west at a speed that is less than the bus’s speed.c. move west at a speed that is equal to the bus’s speed.d. move east at a speed that is equal to the bus’s speed.

Review

• A piece of chalk is dropped by a teacher walking at a speed of 1.5 m/s. From the teacher’s perspective, the chalk appears to fall

a. straight down.b. straight down and forward.c. straight down and backward.d. straight backward.