PhysicsForce and motion, part 2

Image thanks to http://iopscience.iop.org/article/10.1088/0031-9120/48/4/431;jsessionid=5334ED67E8D7EDB2954DFC4FE83D8996.c1

Objectives• By the end of this lesson, you should be able to• Solve 2-D motion problems graphically• Solve 2-D motion problems algebraically

Standards• SP1.

Students will analyze the relationships betweenforce, mass, gravity, and the motion ofobjects.

Scalars and vectors• If a quantity has no direction, it is called a scalar quantity • Examples of scalar quantities • Distance• Speed

• Examples of vector quantities • Displacement • Velocity • Acceleration

“Aren’t you just a ray of sunshine?”• You may remember from geometry that a ray is a fixed point

connected to an unlimited number of points in one direction• The symbol was a little arrow over two known points in the ray• For example: a ray starting at point A and moving through point B

would be shown as• In physics, rays help us to differentiate between scalar and vector

quantities

Instructional video clips:• What is a vector: https://

www.youtube.com/watch?v=FkO6vyMqo8E&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj• Vector notation: https://

www.youtube.com/watch?v=iG-UyKjSaMI&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj&index=2• Components and magnitude of a vector: https://

www.youtube.com/watch?v=f5Mj3_qjyg0&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj&index=3• Finding components of a vector: https://

www.youtube.com/watch?v=oFi6aUAxdMo&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj&index=4

Some more clips• Tip to toe method: https://www.youtube.com/watch?v=_

LrDgGhCTJU&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj&index=6• Adding vectors numerically: https://

www.youtube.com/watch?v=WsV3Qx8KP08&index=7&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj• Adding vectors numerically, part 2: https://

www.youtube.com/watch?v=Rai_yHeByk0&index=8&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj

How to write vectors• Vectors can be used to graphically show how different displacements,

velocities, and forces connect together• Vector quantities have a quantity (how much it is) and a direction

(which way it is) • Each vector amount is assigned a letter (such as A, B, C, etc)• When writing it, it is usually written one of two ways:

• A capital letter with a ray: A

• A emboldened capital letter: A

Positive Attitude, a reminder • When giving directions , you could use • Cardinal directions (north, south, northeast, etc)• Angular direction (in degrees or radians)• With positive meaning one direction and negative meaning the

opposite direction• Commonly, up is positive and down is negative• Commonly, to the right is positive and to the left is negative • Technically, you can make any direction you want as positive (as long as you

are consistent)

North of West• Often, a direction is given between two cardinal directions• For example: 25° North of West• In this example, you would start on the West direction (180° ) and then

move up 25 °• Alternatively, this direction is the same as 65 ° west of North (because 90° -

25° = 65° )• That would mean starting at the north direction (90° ) and moving 65°

degrees down

Graphic representation of vectors• To graphically represent vectors, you must first decide upon a scale• The scale must be such that the vector(s) will fit on your paper

• For example vector B: 250 m @ 17°• I can pick whatever scale I want, but I’ll choose 1 cm = 50 m• Remembering that I may draw lines between 0° and 359° , I am ready

to draw

“Are you just being a negative Nancy today?”• Using a Cartesian coordinate system, we start at the positive x-axis

and move counterclockwise until we come back to where we started

• What does a negative angle mean?• It means that we start at the positive x-axis and move clockwise

• It doesn’t matter (really) which one you use as long as you keep track of the quadrant…more on that later

• Look at the following slide for an example

POSITIVE ANGLES

0°, 360°180°

90°

270°

NEGATIVE ANGLES

0°, -360°-180°

-270°

-90°

What angle is this?

30°

270°

If we measure normally, we would

get 90+30 = 120°

If we measure the negative angle, we would

get -180 + -60 = -240° Remember: They represent the SAME angle!

Ah, the good old days• You should have learned in Algebra about the where angles fall in a

Cartesian coordinate system• The origin is at coordinate (0,0)• The origin is where the first vector is drawn from

“1-2 Step”• One person starts from a point, walks 20 feet forward and then 10

feet to the right• A second person starts from the same point, walks 10 feet to the right

and then 20 feet forward• They both end up at the same location• When adding vectors, it doesn’t matter the order that they are added

Tip to tail• When adding vectors, we use the “tip-to-tail” method• After the first vector is drawn, the next vector starts where the previous one

ended• That vector travels the magnitude and direction as if the tip of the previous

was the origin• This continues until all vectors are on the paper

• Then, a Resultant vector is drawn

Resultant Vector• The “answer” or the result of adding all of the vectors is called the

Resultant Vector, represented by a R or a R

• What it basically shows is where you would end up if all vectors were combined together

• The resultant vector has its own distance and direction

Example problem, graphically• On a previous example, I could change everything to vector form• A = 20 ft @ 90° (aka “forward”)• B = 10 ft @ 0° (aka “to the right”)• I’ll choose 1 cm = 10 ft Notice, that it didn’t matter

which order you added them inOnce measured, the approximate resultant would be:R: 2.2cm @ 27° For my answer, it would have to use my scale to convertR: 22 ft @ 27°

“I can only imagine”• As you might imagine, to get the best graphical answer, you need to do

the following:• Make your scale that is easy to convert from and to• Draw the lines as accurately to the distance as possible• Draw the lines as accurately to the angle as possible• Draw your lines as straight as possible• Make sure that the tip of one vector meets the tail of the next

What is giveth can be taketh away• Some problems may ask to subtract vectors• Consider the following:• A – B = A + (-B)

• We know that a negative answer just means the opposite direction• A + (-B) means that you add vector B AFTER you rotate its direction 180

• If a problem needs you to multiply or divide a vector• Multiply or divide the magnitude (amount) but leave the direction the same

Love/hate relationship• Many students dislike the graphical method of solving physics problems• It can be tedious• It has a fair amount of room for error• Etc

• However, it does have its uses

• In the meantime, let us learn the mathematical way to solve these same problems

Broken down into parts

Look at this vector. We’ll call it CThe coordinates of the tip of C is (3,4)That means that the x-part of C is 3 and the y-part of C is 4So, I can break C down into partsBut I can’t just add them together

C ≠ Cx + Cy

Pythagoras to the rescue• You may remember the Pythagorean Theorem from math class:• A2 + B2 = C2

• Where A and B where two of the sides and C was the hypotenuse

• On our previous drawing, the hypotenuse was C and A and B were Cx and Cy• Substituting, we get• C2 = Cx

2 + Cy2

“Sine, sine, everywhere a sine”

You also learned in Algebra, that, in a right triangle, you can determine distances and angles using trigonometry

sin θ= opposite/hypotenusecos θ= adjacent over hypotenuse

θ

Well, at least I know ahead of time• We know that the hypotenuse is C and the opposite is Cy

• Let’s rewrite the first equation• sin θ = Cy/C (notice that I didn’t bold them because I am only using their

magnitudes)• Solving for Cy: Cy = C sin θ

• Working the same magic with cosine gives• Cx = C cos theta

• Spoiler alert: Breaking vector units down this way is a necessary skill for the rest of the course

Being short isn’t bad• In short:• To find the x-component, use cosine• To find the y-component, use sine

Previous example done mathematically• A: 20 ft @ 90°• B: 10 ft @ 0°

• Broken down:• Ax: 20 cos 90° = 0• Bx: 10 cos 0° = 10• Ay: 20sin 90° = 20• By: 10 sin 0° = 0

Continued• Combine the x-components together and the y-components together• x: 10 + 0 = 10• y: 0 + 20 = 20

• Use the new distance formula to determine the magnitude of R• R = (10)2 + (20)2 = 100 + 400 = 500 = 22.36

• And now for the direction of R

Going off on a tangent• Now that you know the hypothesis, you can use any of the three basic trig

functions to solve for the angle• I have seen that tangent is often used the most• tan θ= 20/10• θ= tan -1 2 = 63.4°

• For the example, R = 22.36 feet @ 63.4°

Divide and conquer• With 1-D motion problems, all movement was along the x-axis or the

y-axis• With 2-D motion problems, movement occurs in both the x-axis AND

the y-axis• Problems will often have to be divided into parts (for both x-axis and

y-axis)

Ignorance is bliss• In almost all cases, you will be told to ignore air resistance

• Since there is nothing resisting motion, acceleration along the x-axis is usually 0.• That means that distance travelled along the x-axis is dependent on constant

velocity along the x-axis and the time required to reach the ground along the y-axis

• So, it is recommended that you determine the y-axis movement first

Classic Physics Thought Experiment• In Florida, several years ago, I monkey escaped and was on the loose in a

city.• Pretend that you are given a tranquilizer gun and told to bring in the

monkey• The monkey is smart and will drop the instant that you pull the trigger• You are smarter than the monkey, and you realize what the monkey will do• At a distance from the monkey, it is time to shoot• Where should you aim: at the monkey, above the monkey, or below the monkey?

Monkey in the middle• Look at this clip from Harvard Natural Sciences Lecture

Demonstrations:• https://www.youtube.com/watch?v=0jGZnMf3rPo• And these two from Mythbusters:• The setup:

http://www.discovery.com/tv-shows/mythbusters/videos/dropped-vs-fired-bullet/ • The finale: https://www.youtube.com/watch?v=D9wQVIEdKh8

The takeaway• What you should take away from the two video sets is this:

• Regardless of how fast an object is travelling horizontally, gravity pulls it independently down

• With no air resistance, any two objects dropped from the same height will hit at the same time.

Example #1• A slingshot fires a shot at 25 m/s at a 0 angle. If the slingshot is 1.5 m

above the ground when fired, how far will the missile go?• Step 1: Determine how long the shot is in the air• Since it is fired at a 0° angle, there is no initial velocity in the y-axis and

acceleration is g• vi = 0 m/s• a = 9.8 m/s2

• y = 1.5 m

• We’ll use: y = vit + ½ at2

Example #1, continued• 1.5 = 0 + ½ (9.81)t2 [Note: I chose 9.81 instead of -9.81 since the shot would be

moving in the same direction]• 1.5 = 4.905t2

• 1.5/4.905 = t2

• .306 = t2

• .55 = t

• Step 2: Determine how far (horizontally) the shot went• vi = 25m/s• t = .55s• a = 0 m/s2 [Since there is no air resistance]• We could use the formula from Step 1, however, it will reduce to

• x = vi t• x = (25)(.55) = 13.75 m

Parabolic motion• If a projectile moves at an angle relative to the ground, the motion will

follow a parabolic shape• It will travel upwards until it temporarily stops and then move down• You will usually be breaking these problems down into three parts:• Movement upwards• Movement downwards• Movement forwards

• Reference the link: http://sdsu-physics.org/physics180/physics180A/units/unit1/chapter3.html

Example #2• An arrow is fired at a velocity of 50 m/s at an angle of 30 relative to the

ground• How high up does it fly?• How long is it in the air for?• How far does it go?• The velocity isn’t completely up nor is it completely forward• We must break into components:• vx = 50 cos 30 = 43.3 m/s• vy = 50 sin 30 = 25 m/s

Example 2, continued• Phase 1: The travel upward (all information is about vertical

movement)• vi = 25 m/s• vf = 0 m/s (at the top of the parabola, it stops moving upward)• a = - 9.81 m/s2 (gravity is in the opposite direction of travel)• y = ?• I can determine the time first or the height first

Example 2, continued, Time first• Regardless of height, the velocity is changing with a constant

acceleration• I will use a = vf – vi / t• -9.81 = (0 – 25)/t• -9.81 = -25/t• -9.81t = -25• t = 2.55 s

• vf2 = vi

2 + 2ay• 0 = 252 + 2(-9.81)y• 0 = 625 – 19.62y• -625 = -19.62y• y = 31.86 m

Example 2, continued, Distance first

Example 2, Phase 2• Now we need to determine the total time in the air• We have have already determined time on the way up is 2.55s and the

height reached is 31.86m• Now, how long it took to fall from that height• vi = 0 m/s (because at the start of the fall, it wasn’t moving downward)• a = 9.81 m/s2 (because arrow is moving same direction as gravity)• y = 31.86m

Example 2, Phase 2, continued• y = vit + ½ at2

• 31.86 = 0 + ½ (9.81)t2

• 31.86 = 4.905t2

• 31.86/4.905 = t2

• 6.5 = t2

• 2.55 s = t

• SURPRISE: the time up is equal to the time down! This is a tip that will help you in future problems• What about velocity?

• vf2 = vi

2 + 2ay• vf

2 = 0 + 2(9.81)(31.86)• vf

2 = 625.09• vf =25 m/s

• Velocity up is the same as velocity right before it hits the ground• CAUTION: The “same time up and down” and the “same velocity up and down” is only true if

the object has no other acceleration than gravity

Example, Phase 3• Total time of flight = 2.55 s (up) + 2.55 s (down) = 5.10 s (total)• Since we “neglect air resistance”, there is no horizontal acceleration• vi = 43.3 m/s• a = 0 m/s2• t = 5.10 s• x = ?• Since a = 0, then we can just use• v = x/t x = vt = (43.3)(5.10) = 220.83 m

More video links• Physics - Mechanics: Motion In Two-Dimension (1 of 21)

Independent Motion in x and y: https://www.youtube.com/watch?v=ld1FLVqJKoU&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj&index=43 • Part 2: https://

www.youtube.com/watch?v=yjjcHxvQ7gY&index=44&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj• Part 3: https://

www.youtube.com/watch?v=W6ZiyhFmk90&index=45&list=PLX2gX-ftPVXVCw9WxxEA4yD14k8yskTSj

Negative angles, take 2• Depending on the direction of different components, you may get a

negative angle• But what does that mean?• It means that angle below the x-axis

• Consider: this can happen in two different places

• To decide which place is the correct one, you need to look at the components

QUADRANT 1QUADRANT 2

QUADRANT 3 QUADRANT 4

X is positiveY is positive

X is negativeY is positive

X is negativeY is negative

X is positiveY is negative

Quadrant breakdowns• Look at tan θ in each quadrant• Quadrant 1: tan θ = + y/ +x = positive answer• Quadrant 2: tan θ = +y/ -x = negative answer• Quadrant 3: tan θ = - y/ -x = positive answer• Quadrant 4: tan θ = - y/ x = negative answer

• So, you could get an answer positive or a negative answer in two different places• Remember, to decide which is which, look at the actual components

For those who like Khan Academy, here are some to look at

• https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/visualizing-vectors-in-2-dimensions• https://

www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/projectile-at-an-angle• https://

www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/different-way-to-determine-time-in-air• https://

www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/horizontally-launched-projectile• https://

www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/total-displacement-for-projectile