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##### Chapter 3: Position, Speed and Velocity

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**Chapter 3: Position, Speed and Velocity**3.1 Space and Position 3.2 Graphs of Speed and Velocity 3.3 Working with Equations**Chapter Objectives**• Calculate time, distance, or speed when given two of the three values. • Solve an equation for any of its variables. • Use and interpret positive and negative values for velocity and position. • Describe the relationship between three-dimensional and one-dimensional systems. • Draw and interpret graphs of experimental data, including velocity versus position, and speed versus time. • Use a graphical model to make predictions that can be tested by experiments. • Derive an algebraic model from a graphical model and vice versa. • Determine velocity from the slope of a position versus time graph. • Determine distance from the area under a velocity versus time graph.**average speed**constant speed coordinates coordinate system displacement instantaneous speed instantaneous velocity origin position rate slope time vector velocity Chapter Vocabulary**Inv 3.1 Position, Speed, and Velocity**Investigation Key Question: How are position, speed, and velocity related?**3.1 Space and position**In physics, the word positionrefers to the location of an object at one instant. A position is always specified relative to an origin. The net change in position relative to the origin is called displacement.**3.1 Position and distance**• Distance is related to, but different from, position. • Distance is a measure of length without regard to direction.**3.1 Position in three dimensions**• Space is three dimensional, so position must also be a three-dimensional variable. • Any position in space can be precisely specified with three numbers called coordinates.**3.1 Positive and negative**• Allowing x, y, and z to have positive and negative values allows coordinates to locate any position in all of space.**3.1 One dimensional problems**• In three-dimensional space, position is a vector. • A vector is a variable that contains all three coordinate values. • Motion in a straight line is easiest to analyze because it is one dimensional. • However, even in one dimension there is an origin and positive and negative values are possible.**3.1 Speed and distance**• Speed is the rate at which distance changes. • In physics, the word ratemeans the ratio of how much something changes divided by how long the change takes. • Constant speed means the same change in distance is traveled every second.**3.1 Calculating speed**• The change in position is a distance traveled in a given amount of time. • To calculate the speed of an object, you need to know two things: • the distance traveled by the object • the time it took to travel the distance**3.1 Calculating speed**• Since speed is a ratio of distance over time, the units for speed are a ratio of distance units over time units.**Calculating speed in meters per second**• You are asked for speed in m/s. • You are given distance = 50 m; time = 7.5 s • Use v = d ÷ t • Plug in values and solve. v = 50 m ÷ 7.5 s ≈ 6.67 m/s • A bird is observed to fly 50 meters in 7.5 seconds. Calculate the speed of the bird in m/sec.**3.1 The velocity vector**• The velocity of an object tells you both its speed and its direction of motion. • A velocity can be positive or negative. • The positive or negative sign for velocity is based on the calculation of a change in position. Two cars going opposite directions have the same speed, but their velocities are different—one is positive and the other is negative.**3.1 The velocity vector**• Velocity is the change in position divided by the change in time.**Chapter 3: Position, Speed and Velocity**3.1 Space and Position 3.2 Graphs of Speed and Velocity 3.3 Working with Equations**Inv 3.2 Position, Velocity, and Time Graphs**Investigation Key Question: How are graphs used to describe motion?**3.2 Graphs of Speed and Velocity**• There are many graphs involving the terms speed, velocity, distance, position, displacement and time. • A position versus time graph shows the details of the actual motion during the trip.**3.2 Average vs. instantaneous speed**• Average speed is the total distance traveled divided by the total time taken. • Instantaneous speed is the apparent speed at any moment, such as on a speedometer.**Interpreting a distance versustime graph**This distance versus time graph shows a boat traveling through a long canal. The boat has to stop at locks for changes in water level. • How many stops does it make? • What is the boat’s average speed for the whole trip? • What is the highest speed the boat reaches?**Interpreting a distance versustime graph**• The boat makes three stops because there are three horizontal sections on the graph. • The average speed is 10 km/h (100 km ÷ 10 h). • The highest speed is 20 km/h. The position changes by 20 km in one hour for the first, third, and fifth hours of the trip.**3.2 Slope**The slopeof a line is the ratio of the “rise” (vertical change) to the “run”(horizontal change) of the line.**3.2 Positive and negative velocities**• When the direction of motion is part of the calculation, changes in position are referred to as displacement.**3.2 Positive and negative velocities**• Average velocity uses the values of displacement and elapsed time from the position vs. time graph. • The average velocity at C is 12 mph.**3.2 Positive and negative velocities**• The slope of the position vs. time graph at any one time is called instantaneous velocity.**Velocity (v) is calculated by dividing the change in**position (Δx) by the change in time (Δt). 3.2 Velocity Equations**3.2 The velocity versus time graph**• The velocity versus time graph has velocity on the y-axis and time on the x-axis. • On this graph, a constant velocity is a straight horizontal line. • Information about an object’s position is also present in the velocity versus time graph.**3.3 Constant Velocity**• This graph shows that the velocity: • is 1 m/s. • stays constant at 1 m/s for 10 seconds.**3.2 The velocity vs. time graph**• The area on a velocity versus time graph is equal to the distance traveled.**A velocity versus time graph can show positive and negative**velocities. 3.2 Relating v vs. t**The position versus time graph, can yield the same**information using the slope to calculate velocity at corresponding time intervals. 3.2 Relating x vs. t**Chapter 3: Position, Speed and Velocity**3.1 Space and Position 3.2 Graphs of Speed and Velocity 3.3 Working with Equations**Inv 3.3 Equations of Motion**Investigation Key Question: How are equations used in physics?**3.3 Working with Equations**An equation is a much more powerful form of model than a graph. While graphs are limited to two variables, equations can have many variables and can be used over a wide range of values.**3.3 Working with Equations**Equations can also be rearranged to show how any one variable depends on all the others.**Calculating time from speedand distance**• You are asked for distance. • You are given time in h and speed in km/h. • Use d = vt. • Solve. d = 2 h × 100 km/h = 200 km • How far do you go if you drive for 2 h at a speed of 100 km/h?**3.3 Solving an equation**• To “solve” means to get a desired variable by itself on one side of an equals sign. • Whatever you do to the left of the equals sign you must do exactly the same to the right. • Get in the habit of solving an equation before you plug in numbers. • More complex problems require you to substitute whole equations for single variables.**3.3 Solving an equation**• To solve this equation for distance (d): • Multiply both sides of the equation by “t”. • Multiplying by “t”on both sides of the equation allows you to cancel a t from the numerator and the denominator on the right side of the equation.**3.3 Position vs. time equation**• The equation says your position, x, is equal to the position you started at, x0, plus the additional amount you traveled, vt.**Calculating time from speedand distance**• A car moving in a straight line at constant velocity starts at a position of 10 meters and finishes at 30 meters in five seconds. What is the velocity of the car? • You are asked for velocity. • You are given that the motion is at constant velocity, two positions, and the time. • Use x = x0+ vt, solve for v. • x – x0 = vt • x – x0 = v t • Substitute numbers for variables: v = 30 m – 10 m = 4 m/s 5 s**3.3 Relating equations and graphs**• In science and engineering, any two variables can be used in the equation for a line, not just x and y.**3.3 Relating equations and graphs**• The y corresponds to x, the position at any time; • the x corresponds totime “t” ; • the slope, m, corresponds to the velocity, v; • the y-intercept, b, corresponds to the initial position, x0.**3.3 Scientific process**• The process of developing a model or theory in science starts with actual experiments and data, and produces a validated model in the form of an equation.**3.3 How to solve physics problems**• Step 1 • Identify clearly what the problem is asking. • Step 2 • Identify the information you are given. • Step 3 • Identify relationships. • Step 4 • Combine the given information and the relationships.