Relationships between Position vs Time and Velocity vs Time graphs

Position vs Time Graph Slope = v = Δd/Δt v = (40 m)/(8 s) v = 5.0 m/s The slope of a position vs time graph equal to the objects velocity. Δd Δt The line of this graph is straight which means the velocity of the object must be constant. Slope = v = Δd/Δt v = (40 m)/(8 s) v = 5.0 m/s How would this constant velocity be shown on a Velocity vs Time graph?

Velocity vs Time Graph The line is horizontal. Why is this? What does a horizontal line on a velocity vs time graph mean? What does the slope of a velocity vs time represent? Answer: The objects acceleration! Since the object is travelling at a constant velocity of 5.0 m/s, it’s acceleration would be zero! The slope of this line is also zero! The two ideas both agree with each other!

Position & Velocity vs Time Graph Constant or Uniform velocity of 5.0 m/s Both of these graph show the SAME motion. They just show it in different ways because the graphs are different!

Position vs Time Graph Velocity vs Time Graph A car travels at a constant speed while covering a distance of 20 m in 10.0 s. The car stops and waits 30 s for a traffic light to change. The car then travels another 50 m in 15.o s before coming to a stop in front of a stop sign. After 5.0 s the cars turns around and heads back towards its original position at a constant speed, reaching this position in 10.0 s. Graph this car’s motion on the Displacement vs Time graph given.

Position vs Time Graph Not moving at all Moving quickly, rightward (+), constant velocity Moving quickly, leftward(-), constant velocity Moving slow, rightward (+), constant velocity

Velocity vs Time Graph Draw a Velocity vs Time graph for the described earlier.

Position vs Time Graph Velocity vs Time Graph These two graphs show the exact same motion. Area under line = displacement = 20 m Slope = speed =2.0 m/s Although they look very different they give you the same information about the car’s position and speed at different times.

Position vs Time Graph Velocity vs Time Graph How is constant velocity shown on a d vs t graph? At an angle How is constant velocity shown on a v vs t graph? A straight horizontal line What is the slope of d vs t graph represent? The velocity What is the slope of v vs t graph represent? The Acceleration

How do you get the speed of an object from a d vs t graph? Analyzing the slope How do you get the speed of an object from a v vs t graph? Analyzing the straight lines How do you get displacement from a d vs t graph? Take two points and subtract df from di How do you get displacement from a v vs t graph? Take the area under the graph between two points

Position vs Time Graph Velocity vs Time Graph Now try this: You leave your home and slowly walk at a constant speed towards a mailbox to mail a letter. On the way to the mailbox you stop to talk to your friend. After a short time you run to the mailbox at a higher constant speed to beat the mail truck. You spend a short time at the mailbox mailing your letter. You begin to stroll home slowly when all of a sudden your are knocked down on the ground by your friendly neighbourhood St. Bernard dog. After a few seconds of face licking you realize you are late for dinner and you really run all the way home. Sketch a Displacement vs Time graph of this motion and then sketch a velocity vs Time graph of this motion.

Position vs Time Graph Velocity vs Time Graph You leave your home and slowly walk at a constant speed towards a mailbox to mail a letter. On the way to the mailbox you stop to talk to your friend. After a short time you run to the mailbox at a higher constant speed to beat the mail truck. You spend a short time at the mailbox mailing your letter. You begin to stroll home slowly When all of a sudden your are knocked down on the ground by your friendly neighbourhood St. Bernard dog. After a few seconds of face licking you realize you are late for dinner and you really run all the way home. D v t t

Position vs Time during Acceleration Notice on a d vs t graph during acceleration it is a curve.

Position vs Time during Acceleration Negative Velocity (speeding up) Negative Acceleration Positive Velocity (slowing down) Negative Acceleration Sad Face = Negative Acceleration Negative Velocity (slowing down) Positive Acceleration Positive Velocity (speeding up) Positive Acceleration https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Module_4_--_Graphing_Motion_and_Acceleration_vs._Deceleration Happy Face = Positive Acceleration

Between t = 0 s and t = 2. 0 s the object moves 8 Between t = 0 s and t = 2.0 s the object moves 8.0 m and its average velocity for this time interval is 4.0 m/s. (vavg = 8.0m/2.0s) Between t = 0 s and t = 4.0 s the object moves 32 m and its average velocity for this time interval is 8.0 m/s. (vavg = 32m/4.0s) Different average velocities depending upon the time interval. Pg41 Physics Serway/Faugh (Holt)

To Find Instantaneous velocity from the Position-Time graph Construct a straight line that is tangent to the position-versus- time graph at the point or instant in question. Calculate the slope of that line.

Velocity vs Time Positive vs Negative Recall velocity is a vector, therefore the direction it is moving will tell us if it has a positive velocity or a negative velocity. An object is moving in the positive direction if the line is located in the positive region of the graph (whether it is sloping up or sloping down). An object is moving in the negative direction if the line is located in the negative region of the graph (whether it is sloping up or sloping down). If a line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

Velocity vs Time Positive vs Negative

Velocity vs Time Speeding up vs Slowing down Speeding up means that the magnitude (or numerical value) of the velocity is getting large. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to - 9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign or direction) is increasing; the speed is getting bigger. The object is speeding up if the line is getting further away from the x-axis (the 0-velocity point); and conversely, if the line is approaching the x-axis, then the object is slowing down.

Velocity vs Time Speeding up vs Slowing down

Position vs Time Graph Velocity vs Time Graph For d vs. t graphs: Slope = rise = displacement = velocity run time For v vs. t graph: Slope = rise = velocity = acceleration run time Area under graph = velocity x time = displacement

Check your understanding

Answers Time 1 Time 2 Time 3 Time 4 Time 5 Time 6 Time 7 Time 8 Position + − Velocity Acceleration Acceleration/ Deceleration A D

Some quite fun practice http://www.physicsclassroom.com/morehelp/grap hs

Draw some Graphs 1) A cycler is at a stop light. When the light turns green it accelerates forward at a constant rate. It then approaches a red light where it slows down at a constant rate and comes to a stop. The cyclist moved forward for 20 seconds, and took 10 seconds to slow down to a stop. Sketch ‘d vs. t’, ‘v vs. t’ graphs of its motion from the start of the green light to the end at the red light.

Draw some Graphs 2) A racing car opens its parachute 30 metres from its stopping point. Upon opening its parachute, it slows down at a constant rate and comes to a stop within 40 seconds. Sketch ‘d vs. t’, ‘v vs. t’ graphs of its motion from the moment it opens its parachute to the point where it comes to a stop.

Draw some Graphs 3) Imagine a car at a stop light. When the light turns green it accelerates forward for 30 metres. From 0 m – 5 m it’s velocity went from 0 m/s to 20 m/s. It remained at 20 m/s for another 20 metres, where it slowed down at a constant rate to a stop over the remaining distance. Sketch ‘d vs. t’ and a ‘v vs. t’ graph of its motion from the start of the green light to the end of the next red light.

Draw some graphs 4) Johnny walks his dog 20 metres for 5 minutes at a constant velocity heading west. He then waits for 1 minute while his dog goes to the bathroom. He then turns around and walks at a constant velocity for 8 minutes heading east for 30 metres to meet with his friend. They stay there talking for 5 minutes, then Johnny heads back home at a constant velocity. Sketch ‘d vs. t’ and a ‘v vs. t’ graph of its motion from the start of the green light to the end of the next red light.