1D Kinematics – Displacement, Velocity, & Acceleration

Essential Questions As this snowboarder moves in a graceful arc through the air, the direction of his motion, and the distance between each of his positions and the next, is constantly changing. What language should we use to describe this motion? How can we visually represent the motion? Copyright © 2007, Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Kinematics The classification and comparison of motion the description of how objects move 4 Types of Motion 1D Kinematics Straight-line motion 2D Kinematics Projectile motion Copyright © 2007, Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1D Kinematics Assumptions: (in analyzing objects in linear motion) the motion is in a straight line (vertical or horizontal) not concerned with why an object is in motion- only how it moves. All objects will be treated like particles.

Motion Diagrams Frames of a movie Particle Model  model the object as if all its mass is at a single point Draw a motion diagram of the car above

Motion Diagrams Draw: Compare:

Motion Diagrams Draw motion diagrams for these examples: Compare these diagrams, what does this show? - type of motion What is needed to compare the positions? - a coordinate system

Motion Diagrams Coordinate System  Sign convention Origin of time  Free to choose Positive (+)Negative (-) x-axisx>0, right of originx<0, left of origin y-axisy>0, above the originy<0, below the origin

Position the location of a particle from a reference point. Find the position: car, cow, and Sue

Displacement The change in the position of an object as it moves from initial position x i to final position x f is its displacement ∆x = x f – x i. Displacement is a signed quantity. Can be either positive or negative.

Clicker 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

Clicker Understanding Samantha starts at a positive position along the x-axis. She then undergoes a negative displacement. Her final position A. Is positive. B. Is negative. C. Could be either positive or negative.

Vector A quantity that requires both a magnitude (or size) and a direction. can be represented by a vector. Graphically, we represent a vector by an arrow.  The velocity of this car is 100 m/s (magnitude) to the left (direction).  Magnitude – size or length of vector Tip – Vectors are clear as MUD – Magnitude, Unit, Direction This boy pushes on his friend with a force of 25 N to the right.

Scalar vs. Vector Scalar Quantity Numbers only (size - magnitude) Examples: 15 seconds, 5 miles States "how much" Expressed by a number and a unit. Can be +, -, or 0.  Distance and Speed  Work and Energy Vector Quantity Size and direction. Magnitude – size or length of vector Examples: Displacement (80 meters north),  Displacement  Velocity  Acceleration  Force  Momentum

Displacement Vectors A displacement vector starts at an object’s initial position and ends at its final position.  It doesn’t matter what the object did in between these two positions. In motion diagrams, the displacement vectors connects successive particle positions.

Distance vs. Displacement Distance: The change in position (without direction).  (magnitude with unit) – scalar  Calculated by finding the path length.  IB uses the symbol s for distance (from the Latin spatium) and S for total distance travelled.  d is also commonly used other places Displacement: the change in position of an object in a stated direction (vector).  (magnitude with unit and direction) – vector  Calculated by finding the difference between the final and the initial positions. ∆x = x f – x i d and s will be interchangeably used.

Motion

Speed rate at which you changed your location  Scalar quantity  Commonly measured in units such as m/s, km/h, or miles per hour (mph). In physics, per is used when we consider the ratio of two quantities. (70 meters per second - 70 m/s) Here per associates the number of units in the numerator (70 m) with one unit of the denominator (1 s).

Speed Calculate the speed of the car and the bicycle The car moves 40 ft in 1 s. The bike moves 20 ft in 1 s.

Uniform Motion Motion of constant speed  Draw all the other displacement vectors for the car and bicycle  Motion diagrams – uniform motion – all the successive frames are uniformly spaced – displacement vectors are identical.

Motion Diagrams Identify the type of motion from the motion diagram

Motion Diagrams Alice is sliding along a smooth, icy road on her sled when she suddenly runs headfirst into a large, very soft snowbank that gradually brings her to a halt. Draw a motion diagram for Alice. Show and label all displacement vectors.

Instantaneous Speed, v Speed at any instant of time. (speedometer) Speed = distance / time (cruise control) Always positive - we have to do calculus or make graphs to determine instantaneous speed!

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

Clicker 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.

Motion – fully characterized What is different about the motion of the two bikes? Motion involves both speed and direction Speed is based on distance traveled and does not measure direction How can we measure direction with position?  Use displacement

Velocity Speed with an indicated direction.  the vector quantity for the scalar quantity speed In the language of physics: Velocity = change in displacement change in time Average Velocity If you change speed or direction, you will change your velocity. Instantaneous Velocity Calculus!

Speed vs. Velocity In the language of physics: Velocity has direction (vector) speed has no direction (scalar). Velocity = change in displacement change in time Speed = change in distance change in time

Velocity Vectors Motion Diagrams  Object’s velocity vector points in the same direction of the displacement vector.  Same time between each dot  Velocity vector is the same as the displacement vector Future, label only velocity vectors on motion diagrams

Problem Solving - Prepare

Problem Solving - Solve

Problem Solving - Assess

Speed and velocity problems Keys: Draw a picture of what is going on (motion diagrams) when appropriate Identify the knowns and unknowns Write the formula you are going to use. Put the right numbers in the formula with units. Show your answer with units! Example: A high school athlete runs 1.00 x 10 2 m in s. What is her speed in m/s? Knowns: Unknown: velocity (speed) Formula: Substitute: Solve: s = 1.00 x 10 2 m t = s v = 8.20 m/s

Speed and velocity problems The high-speed train from Paris to Lyons travels at an average speed of +227 km/h. The trip takes 2.00 hours. How far is Lyons from Paris?

Speed and velocity problems Phil Fiziks is driving down a street at 55 km/h. Suddenly a puppy runs into the street 20 m in front of him. It takes Phil 0.75 seconds to react and apply the brakes. How many meters will he move before the car begins to slow?

Bringing It All Together Matt is traveling along a straight interstate highway. He notices that the mile marker reads 260. He continues to travel until he reaches the 150 mile marker and then retraces his path to the 175 mile marker. Draw a motion diagram of his path. What is his distance traveled? What is his displacement from the 260 mile marker? If he accomplishes the trip in 90 minutes, what is his average speed in miles per hour? His average velocity in miles per hour?

Review The change in distance of an object divided by the time interval over which that change took place is ___________ A change in displacement divided by time gives us ___________

Motion Representations Motion diagram (student walking to school) Table of data Position Time Graph

Position Time Graphs The amount of time taken between readings determines the distance the object moves. time would be the independent variable and be plotted on the x-axis of a graph. Position would be the dependent variable and be plotted on the y-axis of a graph. Position versus Time Time (h)Position (km)

Our graph is a straight line. This tells us that our speed is constant. To determine our average speed we can simple find the slope of the line. Slope = Since out graph is a straight line, we could choose any point on that line to find our values for rise and run. Regardless of the point chosen, the slope would be the same. CONSTANT VELOCITY!

Position Time Graphs Motion Diagrams to Position Time Graphs

Interpreting Position Time Graphs

Constant Velocity – Uniform Motion If the average velocity of an object remains the same for all time intervals, then the object moves at constant velocity. For all intervals, no matter how long or short, the ratio of  d/  t is the same.

What is velocity? Velocity of an object equals the slope of a position versus time (d vs. t) graph for the object.

Speed & Velocity Graphs What does a position versus time graph that looked like this mean? Find the slope of the line (remember: its rise/run). Slope = Therefore, Velocity = 0 m/s. Our car is not moving!

Interpreting Position Time Graphs Interpret the following graph

Different Constant Velocities

What if the curve of x vs. t is not a straight line? A graph like this indicates that the velocity is changing from one time interval to the next. This means our velocity is no longer constant. In this case, we can use a line drawn tangent to our curved graph to find the instantaneous velocity at any point on the curve.

Once our line is drawn tangent, we simply use rise/run to find the slope of our tangent line to determine the instantaneous velocity.Slope =

Instantaneous Velocity - Position Time Graph

Clicker Understanding Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? A.P and Q have the same velocity at 2 s. B.P and Q have the same velocity at 1 s and 3 s. C.P and Q have the same velocity at 1 s, 2 s, and 3 s. D.P and Q never have the same velocity.

What is Acceleration? Acceleration is a change in a state of motion.

Average Acceleration defined as the change in velocity per unit of time. if velocity changes, then there is acceleration.  change in speed or direction acceleration = change in velocity change in time

Acceleration

Acceleration can be either a positive change in motion or a negative change. If it is a negative change it is called deceleration. If an object changes its speed and/or direction in a positive or negative manner, it is accelerating.

Acceleration What three things in a car can change its acceleration? Brakes Accelerator/gas pedal Steering Wheel

Constant Acceleration the average acceleration and the instantaneous acceleration are equal where t 0 = 0

Velocity Time Graphs Time would be the independent variable and be plotted on the x-axis of a graph. Velocity would be the dependent variable and be plotted on the y-axis of a graph. The area under the curve is equal to the displacement

Constant Velocity

Stoplight

Motion Diagram  Velocity Time Graph

Velocity Time Graph

Velocity Time Graphs To find displacement from a v vs. t graph, find the area under the curve for a given time interval Use formulas for area of a triangle, rectangle, or trapezoid (triangle + rectangle)

Total Displacement from v vs t graph Find the displacement for the following example:

Acceleration  Velocity Time Graph

Acceleration Interpretation

Clicker Understanding These four motion diagrams show the motion of a particle along the x-axis. Rank these motion diagrams by the magnitude of the acceleration. There may be ties.

Clicker Understanding These four motion diagrams show the motion of a particle along the x-axis. Which motion diagrams correspond to a positive acceleration? Which motion diagrams correspond to a negative acceleration?

Acceleration vs Time Graphs Area under the curve is the change in velocity Extension  Acceleration is the derivative of velocity  Integrating acceleration gives velocity (area under the curve)  Slope is Jerk or Jolt (derivative of acceleration) If the object is at rest at time t = 0, what is the velocity of the object at time t = 8.0 s?

Kinematic Graphs

Positive Constant Velocity

Negative Constant Velocity

Positive Velocity Positive Acceleration

Positive Velocity Negative Acceleration

Negative Velocity Positive Acceleration

Negative Velocity Negative Acceleration

Clicker 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.

Clicker Understanding A graph of position versus time for a basketball player moving down the court appears like so: A. B. C. D.

Clicker Understanding A graph of velocity versus time for a hockey puck shot into a goal appears like so: A. B. C. D.

Derivation of the Constant Acceleration Equations 1. Starting with, where t 0 = 0  solving for v f  2. Starting with, where t 0 = 0   setting both equations equal  solving for x 

Derivation of the Constant Acceleration Equations 3. Starting with eq. (2), substitute eq. (1) into it  simplifying  4. Starting with eq. (2),solve eq. (1) for time , substituting in eq. (2)  expanding solving for v f 2 

Constant Acceleration Equations Where, v i or v o = initial vel. v f = final velocity d or x = displacement a = acceleration t = time

Visual Overview A motion diagram A pictorial representation A graphical representation A list of variables (knowns & unknowns)

Example A Saturn V rocket is launched straight up with a constant acceleration of 18 m/s 2. After 150 s, how fast is the rocket moving and how far has it traveled? Solving Acceleration Problems Step 1. Include as many visual overview elements as needed. Step 2. Make a list of your known and unknown variables. Then, Step 3. Refer back to the table to see which equation includes both your known variables AND your unknown variable. 88 m/s

Example A Saturn V rocket is launched straight up with a constant acceleration of 18 m/s 2. After 150 s, how fast is the rocket moving and how far has it traveled?? (v y ) i = a = t = (v y ) f = Solving Acceleration Problems Step 3. Refer back to the table to see which equation includes both your known variables AND your unknown variable. 0 m/s 18 m/s 2 ? 150 s (v y ) f = (v y ) i + at (v y ) f = 0 + (18) 150 = 2700 m/s d = ? d = (v y ) i t+ 1/2at 2 d = 202,500m = 2.0 x 10 5 m d = 0(150) + ½(18)(150) 2

Acceleration Problems An airplane starts from rest and accelerates at a constant m/s 2 for 30.0s before leaving the ground. What is its displacement during this time? Knowns & Unknowns v i = a = t = d = Equation? ? 0 m/s 3.00 m/s s

Acceleration Problems An airplane accelerates from a velocity of 21 m/s at the constant rate of 3.0 m/s 2 over +535 m. What is its final velocity? Knowns & Unknowns d = v i = v f = a = t = Equation? Don’t forget to take the square root of v f 2 since we are trying to find v f and not v f 2 !

Braking Distance A car is traveling at a speed of 30 m/s on wet pavement. The driver sees an obstacle ahead and decides to stop. From this instant it takes him 0.75 s to begin applying the brakes. Once the brakes are applied, the car experiences an acceleration of -6.0 m/s 2. How far does the car travel from the instant the driver notices the obstacle until stopping?

Accelerated Algebra You will be provided with the original equations listed in the table on tests and quizzes, but perhaps the most challenging part will be to correctly solve these for the variable in question. Practice by solving: v f 2 = v i 2 + 2ad for d: v f 2 = v i 2 + 2ad for a: v f = v i + at for t: Rewrite: v f = v i + at when v i = 0. Rewrite: d = v i t + ½ at 2 when v i = 0 Rewrite: v f 2 = v i 2 + 2ad when v i = 0