Inv 3.1 Position, Speed, and Velocity Investigation Key Question: How are position, speed, and velocity related? 1

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.

Inv 3.2 Position, Velocity, and Time Graphs Investigation Key Question: How are graphs used to describe motion? 14

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.