Reference frames and displacement elapses between each image Section 2–1 Reference frames and displacement Motion diagrams A motion diagram is a graphical plot of the position of an object at equal time intervals. A motion diagram is a graphical plot of the position of an object at equal time intervals. A motion diagram is a graphical plot of the position of an object at equal time intervals. A motion diagram is a graphical plot of the position of an object at equal time intervals. The same amount of time elapses between each image and the next.

Particle models A particle model is a simplified version of the motion diagram in which the object in motion is replaced by a series of single points. A particle model is a simplified version of the motion diagram in which the object in motion is replaced by a series of single points. A particle model is a simplified version of the motion diagram in which the object in motion is replaced by a series of single points. A particle model is a simplified version of the motion diagram in which the object in motion is replaced by a series of single points. 1 2 3 4

More examples Motion diagrams can also show accelerated and two dimensional motion. Motion diagrams can also show accelerated and two dimensional motion. Motion diagrams can also show accelerated and two dimensional motion.

Types of motion (a) is translational (b) is rotational and Motion can be along a linear path. This is translational motion. Objects can also rotate or spin. This is called rotational motion. Motion can be along a linear path. This is translational motion. Objects can also rotate or spin. This is called rotational motion. Motion can be along a linear path. This is translational motion. Objects can also rotate or spin. This is called rotational motion. (a) is translational (b) is rotational and translational

“With respect to the earth.” “With respect to the train.” Frames of reference Measurements of position, distance, and speed must be relative to a reference frame or frame of reference. Each must be given an origin and a coordinate system. Measurements of position, distance, and speed must be relative to a reference frame or frame of reference. Each must be given an origin and a coordinate system. Measurements of position, distance, and speed must be relative to a reference frame or frame of reference. Each must be given an origin and a coordinate system. Earth as reference. Speed of train = 80 km/h “With respect to the earth.” Train as reference. Speed of man = 5 km/h “With respect to the train.”

Coordinate systems A coordinate system is needed to define position and time in a motion diagram. A coordinate system is needed to define position and time in a motion diagram. A coordinate system is needed to define position and time in a motion diagram. An origin is usually defined by the observer for time and position set to zero. An origin is usually defined by the observer for time and position set to zero. An origin is usually defined by the observer for time and position set to zero. Subsequent position is then relative to the origin. Subsequent position is then relative to the origin. Subsequent position is then relative to the origin. Distance is a measure of how far a certain position is from the origin (in meters). Distance is a measure of how far a certain position is from the origin (in meters). Distance is a measure of how far a certain position is from the origin (in meters). Distance is a measure of how far a certain position is from the origin (in meters). Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position. Displacement is the distance and direction from the origin, and represents the change in position.

Linear coordinate system Linear coordinate systems are one dimensional. Position for an object is plotted on a line relative to an origin (x0=0). Linear coordinate systems are one dimensional. Position for an object is plotted on a line relative to an origin (x0=0). x0 x1

Integer position The cow is at position –5 km. The cow is at The post office defines the origin, or zero position of the coordinate system. The cow is at position –5 km. The cow is at position –5 km. The car is at position +4 km. The car is at position +4 km.

Other linear coordinate diagrams Linear coordinate systems do not have to be horizontal. Here are some other examples. Linear coordinate systems do not have to be horizontal. Here are some other examples.

Plane coordinate systems For two dimensional motion on a plane, we use the standard Cartesian coordinate axes: For two dimensional motion on a plane, we use the standard Cartesian coordinate axes: –x

Time intervals ∆t = tf – ti Some initial time, not necessarily at the origin, is labeled ti. Some initial time, not necessarily at the origin, is labeled ti. Some initial time, not necessarily at the origin, is labeled ti. A final time measurement is labeled tf. A final time measurement is labeled tf. The time interval is the difference between the final and initial time in seconds. The time interval is the difference between the final and initial time in seconds. The time interval is the difference between the final and initial time in seconds. The time interval is the difference between the final and initial time in seconds. The time interval is the difference between the final and initial time in seconds. The time interval is the difference between the final and initial time in seconds. ∆t = tf – ti

If we are interested in the motion of the car from this position, we assign to this point the time t0 = 0 s. If we’re only interested in the braking part of the motion, we would assign t0 = 0 s here.

Vector and scalar quantities A scalar quantity has size, or magnitude, only. A scalar quantity has size, or magnitude, only. A scalar quantity has size, or magnitude, only. Examples of scalar quantities include distance, time, temperature, work, energy, and mass. Examples of scalar quantities include distance, time, temperature, work, energy, and mass. Vector quantities include both magnitude and direction. Vector quantities include both magnitude and direction. Vector quantities include both magnitude and direction. Examples include displacement, velocity, acceleration, force, and momentum. Examples include displacement, velocity, acceleration, force, and momentum.

Displacement ∆s = sf – si ∆x = xf – xi ∆y = yf – yi Displacement is a vector quantity. Displacement is a vector quantity. It shows the distance and direction from the initial to the final position of an object. It shows the distance and direction from the initial to the final position of an object. It shows the distance and direction from the initial to the final position of an object. It represents the change in position over time. It represents the change in position over time. It represents the change in position over time. Displacement values must include both a magnitude and a direction. Displacement values must include both a magnitude and a direction. Displacement values must include both a magnitude and a direction. ∆s = sf – si ∆x = xf – xi ∆y = yf – yi on a x-coordinate plot on a y-coordinate plot

Positive displacement What is Sam’s displacement? xf = 150 m xi = 50 m ∆x = +100 m Negative displacement What is Sam’s displacement? xf = 0 m xi = 50 m ∆x = –50 m

Displacement vectors Vectors are represented by arrows, or rays. Vector arrows are added by placing the tail of the second arrow on the head of the first. Vector arrows are added by placing the tail of the second arrow on the head of the first. Vector arrows are added by placing the tail of the second arrow on the head of the first. The resultant vector, is the vector sum of all the individual vectors. It goes from the initial to the final position. The resultant vector, is the vector sum of all the individual vectors. It goes from the initial to the final position. The resultant vector, is the vector sum of all the individual vectors. It goes from the initial to the final position. The resultant vector, is the vector sum of all the individual vectors. It goes from the initial to the final position. The resultant vector, is the vector sum of all the individual vectors. It goes from the initial to the final position. resultant Did Jane really walk this way?