Four Types of Motion We ll Study

Four Types of Motion We ll Study

The branch of mechanics that studies the motion of a body without caring about what caused the motion.

Kinematics definitions Kinematics branch of physics; study of motion Position (x) where you are located Distance (d ) how far you have traveled, regardless of direction Displacement ( x) where you are in relation to where you started

Vector versus Scalar A scalar has only a size - no direction Example 20 m/s A vector has a size (magnitude) and a direction Example 20 m/s due North

Motion Motion is just a change in position over time. Measurements needed Distance Time Direction helps

Position Mark a zero point on the line, pick a direction to be positive, and measure from there. Positions can be positive or negative. Units of position: centimeters, meters, kilometers, inches, feet, miles, etc. Common symbol: x

Positions are Relative Different people can mark the line differently, so they can get different numbers for position. The position number (and unit) really don t mean anything until you specify where you marked 0, and which way you made positive - your frame of reference.

Distance The total length of the path traveled by an object. Does not depend upon direction. (scalar) How far have you walked?

Displacement The change in position of an object. Depends only on the initial and final positions, not on path. Includes direction. (vector) How far are you from home?

Distance vs Displacement B 100 m displacement 50 m A distance

Displacement Represented by d d = d f d i -OR- Represented by x. x = x 2 - x 1 where x 2 = final position x 1 = initial position

If displacement is positive, the object moves to the right. If the displacement is negative, the object moves to the left.

Checking Understanding Maria is at position x = 23 m. She then undergoes a displacement x = 50 m. What is her final position? A. 27 m B. 50 m C. 23 m D. 73 m

Answer Maria is at position x = 23 m. She then undergoes a displacement x = 50 m. What is her final position? A. 27 m B. 50 m C. 23 m D. 73 m

Rates A rate measures how fast something changes. In physics, a rate is almost always calculated as a quantity divided by time. Rate Q changes = change in Q time for Q to change

Speed and Velocity

SPEED Speed is the rate position changes, or the rate distance is covered. Speed is telling how fast something is moving. Speed is always a positive number. (Remember the distance vs. time graph) Quantities that can be described by a single number (magnitude) are called scalars.

Measuring Speed Speed is the amount of distance an object travels in a certain amount of time. The equation for speed is Speed = distance time

Here is a little bug located at 0 cm at 0 seconds.

10 seconds later he is at 50 cm

SPEED The speed of the bug is just his distance traveled divided by how long it takes. Average speed = distance/time Speed = 50 cm/10s s = 5 cm/s

The skier has traveled 400 meters in 6 seconds. What is his speed? Distance traveled is The time it took was 6 seconds. The speed is the distance traveled divided by the time taken. speed = 400 meters 6 seconds speed = 66.67 m s

Problems When Evelyn Ashford was in the Olympics she broke the record for the 200 m run by completing it in 11s. What was her speed? Remember: We have a specific 5-step method for solving problems the GUESS method.

Problems The Pacific Plate moves at an pace of 8.1 cm/year. How far does the plate move in a century? G: givens v = 8.1cm/year t = 1 century = 100 years U: unknowns d =? E: equation v = d/t Rearrange by multiplying sides by t v(t) = d (t) t vt =d or d = vt S: substitution d = ( 8.1 cm )(100 yrs ) yr S: solution v = 810 cm, or 8.1 m

In 1931, "'Big Bill' Tilden delivered the fastest serve ever officially measured. The speed was 73.14 m/s. If the serve covered 30.5 m, how much time did Bill s opponent have to react before the ball reached him? G: givens d = 30.5 m v = 73.14 m/s U: unknowns t =? Problems E: equation v = d/t S: substitution t(v) = d v v t = d/v t = 30.5m 73.14m/s S: solution t = 0.42 sec

VELOCITY Velocity is speed and direction. Velocity is how fast and which way. Quantities that have direction are called vectors.

Average velocity = displacement / time Units are meters/second (m/s) Average velocity does not tell you speed or velocity at each moment. Can be positive or negative depending on direction moved. Time can never be negative!

Average -VS- Instantaneous Speed or Velocity Average distance / time Easier to calculate Instantaneous No easy calculation Moment in time or any given instant What's on a cars speedometer On graph study small time interval (slope at that point)

Suppose our bug stars off at 100 cm.

And ends up at 80 cm 10 seconds later.

Velocity = change in position/time Velocity = (final position initial position)/time Velocity = (80cm 100cm)/10 seconds Velocity = - 2 cm/s

A positive velocity means moving to the right. (usually) A negative means moving to the left. (usually)

Practice Heather and Matthew walk eastward with a speed of.98 m/s. If it takes them 34 min to walk to the store, how far have they walked?

Practice Heather and Matthew walk eastward with a speed of.98 m/s. If it takes them 34 min to walk to the store, how far have they walked? Knowns? What do you know? Write it down. Speed =.98 m/s, time = 34 minutes (2040 sec) Unknown? What do you want to know? How far? Distance =? Equation? Write the equation you ll use. Speed = distance / time Work the problem..98 m/s = distance / 2040 sec; d = 1999.2 meters

Position Time Graphs

Position Time Graphs Position of Moving Object over Time Position 12 10 8 6 4 2 Position is not changing over time. Not Moving v =0 0 0 5 10 15 20 25 30 Time

Position Position Time Graphs Position of Moving Object over Time 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 Time v =Δx = slope Δt Position is changing, i.e. object is moving Velocity determined by slope of position-time graph For Linear Relationship (straight line), slope is constant.

Position Time Graphs Position of Moving Object over Time 60 50 Position 40 30 20 10 0 0 5 10 15 20 25 30 Time

Position Position Time Graphs Position of Moving Object over Time 100 80 60 40 20 0 0 5 10 15 20 25 30 Time Line A and Line B have same slopes; Object A and Object B moving with same velocity Line A and Line B have different y- intercepts; Object A and Object B have different starting points.

Position Position Time Graphs Position of Moving Object over Time Slope is constant, velocity is constant 80 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 Time Slope is negative, velocity is negative What does this mean?

Graph of Average vs. Instantaneous Velocity Recall Instantaneous Velocity is velocity at a given instant in time. You can use slope to figure this out

Position (m) Graph of Average and Instantaneous Velocity Slope Remains constant Velocity is constant Position vs. Time 40 35 30 V avg = Δd Δt 25 20 15 10 5 0 0 5 10 15 20 Time (s)

Graph of Average and Instantaneous Velocity(curved is not constan velocity) Slope is not constant and changes so Velocity is not constant Position (m) 35 30 25 20 15 10 5 0 Position vs Time 0 5 10 15 20 25 Time (s) Velocity is the slope of the tangent line at this point.

Motion Diagrams

Making a Motion Diagram

Examples of Motion Diagrams

Motion Diagrams

Motion Diagrams

A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object. The Particle Model

Position and Time The position of an object is located along a coordinate system. At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0). Slide 1-17

Motion Diagrams-Particle a.k.a Oil Drip

Checking Understanding Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

Answer Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster? A

Checking Understanding Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C.They are both moving at the same speed.

Answer Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C.They are both moving at the same speed.

Acceleration

ACCELERATION Acceleration is a change in velocity per time. a = (v f -v i )/t a = Δv/t So an object accelerates if it Speeds up Slows down Or changes the direction it is moving,

Acceleration = change in velocity/time Acceleration = (final velocity initial velocity)/time Acceleration has units of -velocity/time -(meters per second)/seconds -Or m/s/s or m/s 2

General Rule If the sign of the velocity and the sign of the acceleration is the same, the object speeds up. If the sign of the velocity and the sign of the acceleration are different, the object slows down.

Acceleration is a vector because it has size and direction. Ex. If the final velocity is greater than the initial velocity the v will be positive If the final velocity is less than the initial velocity the v will be negative.

confusion Acceleration, like velocity, has direction. For objects moving along a straight line the direction of an object s acceleration is denoted by plus or minus.

Accelerating vs Decelerating A negative acceleration doesn t always mean the object is slowing down. It could be moving in the negative direction.

Constant Acceleration Velocity increases or decreases by exactly the same amount during each time interval. The displacement for each time interval increases or decreases by increasing or decreasing amounts.

5 Kinematic Equations All the kinematic equations are related. They can be derived using the idea that acceleration will be constant. There is always more than one way to solve each problem. In general though, one equation is generally easier to use then others.

KINEMATIC EQUATIONS Let s review the quantities we ve seen so far: The fundamental quantities are displacement (x or y), velocity (v), and acceleration (a). Acceleration is a change in velocity, from an initial velocity (v 0 or v i ) to a final velocity (v or v f ).

KINEMATIC EQUATIONS And finally, the motion takes place during some elapsed time interval, Δt. If we agree to start our clocks at time t i = 0, then we can just write t instead of Δt, which simplifies the notation. Therefore, we have five kinematic quantities: x, v 0, v, a, and t.

Equations - Kinematic a = (v f -v i )/ (t f -t i ) OR v/ t v avg = ½ (v i +v f ) Now using substitutions. d = 1/2(v i +v f ) t v f = v i +a t d f =d i + v i (t f ) + ½ a(t f ) 2 v f2 = v i2 +2a d

KINEMATIC EQUATIONS These five quantities are related by a group of equations that we call the BIG FOUR: BIG FOUR #1: x = ½(v i + v f )t a BIG FOUR #2: v f = v i + at x BIG FOUR #3: x f = x i + v i t + ½at 2 v f BIG FOUR #4: v 2 f = v i2 + 2ax t Variable missing

KINEMATICS BIG FOUR Each of the BIG FOUR equations is missing one of the five fundamental quantities. The way you decide which of equation to use when solving a problem is to determine which of the fundamental quantities is missing from the problem that is, which quantity is neither given nor asked for and then use the equation that doesn t have that variable.

KINEMATICS BIG FOUR For example, if the problem never mentions the final velocity v is neither given nor asked for the equation to use is the one that s missing v f that s BIG FOUR #3 x f = x i + v i t + ½at 2

The equations Tips It can be confusing, but sometimes These equations are written different. In this case x is distance or d. Also, the subscripts i and f have been replaced with o and no subscript for f.

Tips You won t get a problem wrong by using the wrong equation. You ll get no answer at all because a piece of information is missing. The first step is to READ CAREFULLY!

Tips Read slowly, carefully, and more than once. For example, you might gloss over that an object starts at rest and later be confused as to the missing initial velocity. However, when an object starts at rest it is implied that the initial velocity is 0m/s.

tips Know what you re looking for. What is the final velocity? means V=? How far will the car travel before it comes to a stop? means d=?

Tips Choose the correct equation. Choose the equation that best relates to the facts given. If the problem doesn t include time, you probably won t be using an equation using time.

Practice Problems Do on Board A hockey player glides along the ice at a constant speed of 1.25 m/s in the positive direction onto a rough section of ice, which slows him. If he stops in 5.0 s, what is the magnitude and direction of his acceleration?

a = (v f -v i )/ (t f -t i ) OR v/ t a = o.25 m/s 2, negative

Practice Problems Do on Board A race car travels on a racetrack at 44 m/s and slows at a constant rate to a velocity of 22 m/s over 11 s. How far does it move during this time?

d = ½ (v i +v f ) t 363 m

Practice Problems Do on Board Jill jogs at a velocity of 2.50 m/s. If she then accelerates at a constant -0.10 m/s 2, how fast will she be jogging when she has moved 10.0m?

v f2 = v i2 +2a d 21 m/s

Acceleration Practice If a car can go from 0 mi/hr to 60 mi/hr in 8 seconds, what is its acceleration? G: U: E: S: S:

Acceleration Practice A car must come to an emergency stop. What is its acceleration if it took 8 seconds for it to stop when it was traveling at 30 m/s? G: U: E: S: S:

G: U: E: S: S:

G: U: E: S: S:

Velocity Time Graphs

Velocity-Time Graphs Have described motion using Position-Time Graphs. Can also describe motion using Velocity-Time Graphs.

Graphs The slope and shape of a graph describes the object s motion.

Velocity (m/s) Velocity-Time Graphs Constant Velocity Velocity of Moving Object over Time 12 Δd= vδt Displacement = Area of under curve (rectangle) 10 8 6 4 2 0 0 5 10 15 20 25 30 Time

Velocity (m/s) Velocity-Time Graphs Velocity is Not Constant Velocity increasing over time Velocity of Moving Object over Time 7 Δd= Area of under curve (triangle) 6 5 4 3 2 1 0 0 5 10 15 20 25 30 Time

Velocity (m/s) Velocity-Time Graphs Velocity is Not Constant Velocity of Moving Object over Time 9 8 7 6 5 4 3 2 1 0 0 5 10 15 20 25 30 Time Velocity decreasing over time Δd= Area of under curve (triangle)

Velocity (m/s) Velocity-Time Graphs Velocity is Not Constant Velocity of Moving Object over Time 7 How Object Moving? From t 0 to t 30 : Velocity increasing From t 30 to t 60 : Velocity constant From t 60 to t 90 : Velocity decreasing 6 5 4 3 2 1 0 0 10 20 30 40 50 60 70 80 90 100 Time

Pick the constant velocity graph(s) x A v C x B t v t D t t

Checking Understanding Here is a motion diagram of a car moving along a straight stretch of road: Which of the following velocity-versus-time graphs matches this motion diagram? A. B. C. D.

Answer Here is a motion diagram of a car moving along a straight stretch of road: Which of the following velocity-versus-time graphs matches this motion diagram? A. B. C. D.

Checking Understanding A graph of position versus time for a basketball player moving down the court appears like so: Which of the following velocity graphs matches the above position graph? A. B. C. D.

Answer A graph of position versus time for a basketball player moving down the court appears like so: Which of the following velocity graphs matches the above position graph? A. B. C. D.

Checking Understanding A graph of velocity versus time for a hockey puck shot into a goal appears like so: Which of the following position graphs matches the above velocity graph? A. B. C. D.

Answer A graph of velocity versus time for a hockey puck shot into a goal appears like so: Which of the following position graphs matches the above velocity graph? A. B. C. D.

Free Fall Free fall is motion under the influence of gravity only - no friction or air resistance. Gravity accelerates the object toward the earth.

Acceleration in Free Fall The acceleration of an object in free fall is constant. At the surface of Earth, the free-fall acceleration is about 10 m/s 2, or 9.8 m/s 2 if you have a calculator (or 32 ft/s 2 or 22 mi/hr/s in English units). g = 9.8 m/s 2 downward. a = -g if up is positive. acceleration is down when ball is thrown up EVERYWHERE in the balls flight.

Summary v = v - gt o x = x + v t - 1/2 gt 2 o o v 2 = v 2g( x) o2

Symmetry When something is thrown upward and returns to the thrower, this is very symmetric. The object spends half its time traveling up; half traveling down. Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with. Acceleration is 9.8 m/s 2 everywhere!

Air Resistance The effect of air resistance is to slow an object down and/or decrease its acceleration.

Gravity Problems At what speed would a penny hit the water in a wishing well if it falls for 3.2 seconds? G: (For v 1 : If I drop something what would its initial velocity be, when it is still in my hands?) U: E: S: S:

Gravity Problems If a water balloon is dropped out of a 2 story window and hits the person below at 14 m/s, how long was it falling? G: U: E: S: S: