Chapter 3: Position, Speed and Velocity 3.1 Space and Position 3.2 Graphs of Speed and Velocity 3.3 Working with Equations 2

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.

Chapter Vocabulary average speed constant speed coordinates coordinate system displacement instantaneous speed instantaneous velocity origin position rate slope time vector velocity

Inv 3.1 Position, Speed, and Velocity Investigation Key Question: How are position, speed, and velocity related? 5

3.1 Space and position In physics, the word position refers 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 rate means 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 A bird is observed to fly 50 meters in 7.5 seconds. Calculate the speed of the bird in m/sec. 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

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 18

Inv 3.2 Position, Velocity, and Time Graphs Investigation Key Question: How are graphs used to describe motion? 19

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 versus time 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 versus time 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 slope of a line is the ratio of the “rise” (vertical change) to the “run”(horizontal change) of the line.

3.2 Speed is the slope of the distance versus time graph

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.

3.2 Velocity Equations Velocity (v) is calculated by dividing the change in position (Δx) by the change in time (Δt).

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.

3.2 Relating v vs. t A velocity versus time graph can show positive and negative velocities.

3.2 Relating x vs. t The position versus time graph, can yield the same information using the slope to calculate velocity at corresponding time intervals.

Chapter 3: Position, Speed and Velocity 3.1 Space and Position 3.2 Graphs of Speed and Velocity 3.3 Working with Equations 37

Inv 3.3 Equations of Motion Investigation Key Question: How are equations used in physics? 38

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 speed and distance How far do you go if you drive for 2 h at a speed of 100 km/h? 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

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 speed and 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 to time “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.

Calculating distance from time and speed A space shuttle is traveling at a speed of 7,700 meters per second. How far in kilometers does the shuttle travel in one hour? At an altitude of 300 kilometers, the circumference of the shuttle’s orbit is 42 million meters. How long does it take the shuttle to go around Earth one time?

Calculating distance from time and speed This is a two-part problem asking for distance in kilometers and time in hours. You are given a speed and time for the first part, and a speed and distance for the second. d = vt, and t = d ÷ v 1 h = 3,600 s 1 km = 1,000 m Part 1: d = (7,700 m/s)(3,600 s) = 27,720,000 m Convert to kilometers: 27,720,000 m ÷ 1,000 km/m = 27,720 km Part 2: t = 42 × 106 m ÷ 7,700 m/s = 5,455 s Convert to minutes: = 5,455s ÷ 60 s/min = 90.9 minutes

Slow Motion Photography A video camera does not photograph moving images. It takes a sequence of still images called frames and changes them fast enough that your brain perceives a moving image. You can use an ordinary video camera to analyze motion in laboratory experiments.