Motion Graphs

Another way to describe motion is to graph displacement and velocity as a function of time it

SLOPE REVIEW

SLOPE REVIEW

SLOPE REVIEW

SLOPE REVIEW

SLOPE REVIEW

Position (m) Rise Run Time (s)

VELOCITY Velocity is represented by the SLOPE of the curve on a displacement vs. time graph. In this class, a positive (+) slope indicates a forward direction and a negative (-) slope indicates a backwards direction (return).

Use the position graph to answer the following: a. What is the object’s velocity from 10 – 15 seconds? 0 m/s (object is at rest) b. What is the object’s velocity from 15 – 25 seconds? = -4 m/s c. What is the object’s velocity from 0 – 40 seconds? = -1 m/s

Consider this trip… Position (m) Time (s)

How fast is the car going at this instant in time? INSTANTANEOUS VELOCITY How fast is the car going at this instant in time? Position (m) Time (s)

INSTANTANEOUS VELOCITY The slope of a curve at a given point, is equal to the slope of a tangent line at that point. Position (m) Time (s)

INSTANTANEOUS VELOCITY The slope of this line represents instantaneous velocity at the indicated point. Position (m) Time (s)

Hint: What is the meaning of the slope of the x(t) graph? Describing Motion The x(t) graph describes a 1-D motion of a train. What must be true about this motion? Speeds up all the time Slows down all the time Speeds up part of the time Slows down part of the time Speeds up & then slows down Slows down & then speeds up Hint: What is the meaning of the slope of the x(t) graph?

Position vs. Time Graphs B t x A t1 The x(t) graph displays motions of two trains A and B on parallel tracks. Which statement is true? At t1 both trains have the same velocity At t1 both trains have the same speed At t1 both trains have the same acceleration Both trains have the same velocity sometime before t1 The trains never have the same velocity

Motion can be described with a velocity vs. time graph.

Velocity vs. Time 1 Velocity (m/s) Time (s) constant acceleration constant velocity Time (s)

Velocity vs. Time 2 + - Velocity (m/s) Time (s) + velocity 0 velocity Time (s) - velocity -

The Slope of Velocity vs. Time Graphs Velocity (m/s) Rise =  v Run =  t Time (s)

Use the velocity graph to answer the following: a. What is the object’s velocity from 4 – 7 seconds? 3 m/s b. What is the object’s acceleration from 4 – 7 seconds? 0 m/s2 (constant velocity) c. What is the object’s acceleration from 2 – 4 seconds? = 1 m/s2

Consider a trip… A train travels at 5 miles per hour for 1 hour. What is its displacement after 1 hour? Displacement can also be determined by finding the area under the curve of a velocity vs. time graph.

Using a graph Velocity (m/s) 5 mi/hr 1 hour Time (hrs)

Find the “area under the curve” (AUC). Velocity (m/s) 5 mi/hr 1 hour Time (s) Area = l x w = 1 hour x 5 mi/hr = 5 miles.

Graphical analysis summary Displacement vs. Time graph: Slope = velocity Velocity vs. Time graph: Slope = acceleration AUC = displacement

Draw a “v vs. t” graph for: A blue car moving at a constant speed of 10 m/s passes a red car that is at rest. This occurs at a stoplight the moment that the light turns green. The clock is reset to 0 seconds and the velocity-time data for both cars is collected and plotted. The red car accelerates from rest at 4 m/s/s for three seconds and then maintains a constant speed. The blue car maintains a constant speed of 10 m/s for the entire 12 seconds.

The answer

QUICK QUIZ 2.3 Parts (a), (b), and (c) of the figure below represent three graphs of the velocities of different objects moving in straight-line paths as functions of time. The possible accelerations of each object as functions of time are shown in parts (d), (e), and (f). Match each velocity-time graph with the acceleration-time graph that best describes the motion.

One way to look at the motion of an object is by using a motion map.

Position and direction Imagine a toy car traveling along a piece of paper and dropping a dot of ink at a given time interval (say 1 drop every second). It could produce a trail that looks like this: Position and direction of object at a given instant in time Same time interval Between dots Time starts at zero distance

Position and direction If the car produced the following motion map, how long did it take the car to travel the length of the paper? distance Same time interval Between dots Position and direction of object at a given instant in time Time starts at zero 1 sec 2 sec 3 sec 4 sec 5 sec 6 sec 7 sec 8 sec 8 seconds total

Position and direction Describe the motion of the car? Stopped, constant velocity, accelerating or decelerating? distance Same time interval Between dots Position and direction of object at a given instant in time Time starts at zero Constant velocity How do you know?

Graphing Examples

a. Draw the motion map for the following: Object accelerates for 3 seconds. Then travels at a constant velocity for 2 seconds Then decelerates for 3 seconds Stops for 2 seconds Then returns to the start in 4 seconds at a constant velocity. b. Sketch the velocity graph for the above motion.