Position-Time Graphs SNC2D – Physics M. M. Couturier

Position-Time Graphs As the term suggests, a position-time graph is a graph where position is plotted on the y-axis and time is plotted on the x-axis. As the term suggests, a position-time graph is a graph where position is plotted on the y-axis and time is plotted on the x-axis.

Position-Time Graphs Since velocity is related to position and time (m/s), the slope of the function will reveal the average velocity. Since velocity is related to position and time (m/s), the slope of the function will reveal the average velocity.

Position-Time Graphs However, to understand the nature of the velocity, we must understand all four kinds of slopes: increasing, decreasing, zero and infinite. However, to understand the nature of the velocity, we must understand all four kinds of slopes: increasing, decreasing, zero and infinite.

Position-Time Graphs This is an example of a zero slope. The displacement here is ∆d = 0 over a period of time, t. The average velocity here is therefore: This is an example of a zero slope. The displacement here is ∆d = 0 over a period of time, t. The average velocity here is therefore: V avg = ∆d = 0 = 0 V avg = ∆d = 0 = 0 ∆t t ∆t t Essentially, the object is at rest. Essentially, the object is at rest.

Position-Time Graphs This is an example of an increasing slope. ∆d and ∆t are both positive and therefore, V avg will also be positive. Also, since it is a straight line (linear), the velocity is constant, meaning the velocity is the same. This is an example of an increasing slope. ∆d and ∆t are both positive and therefore, V avg will also be positive. Also, since it is a straight line (linear), the velocity is constant, meaning the velocity is the same.

Position-Time Graphs This is an example of a decreasing slope. ∆d is negative and since ∆t can only be positive, V avg will be negative. They are either driving towards a reference point or returning to their origin. This is an example of a decreasing slope. ∆d is negative and since ∆t can only be positive, V avg will be negative. They are either driving towards a reference point or returning to their origin.

Position-Time Graphs There is another possibility, but it leaves the realm of Newtonian physics; that of the infinite slope, where we have a ∆d but with a t = 0. This essentially means that an object has gone from one place to another in 0 time. We will not study this situation further. There is another possibility, but it leaves the realm of Newtonian physics; that of the infinite slope, where we have a ∆d but with a t = 0. This essentially means that an object has gone from one place to another in 0 time. We will not study this situation further.

Position-Time Graphs In order to calculate the average velocity using a position-time graph, you need to isolate two points on a single line. Look at the position-time graph below and write down the coordinates of any two points on the line. In order to calculate the average velocity using a position-time graph, you need to isolate two points on a single line. Look at the position-time graph below and write down the coordinates of any two points on the line.

Position-Time Graphs Since the average velocity is defined by: Since the average velocity is defined by: V avg = ∆d V avg = ∆d ∆t ∆t The two points that we have selected provide both ∆d and ∆t. The two points that we have selected provide both ∆d and ∆t.

Position-Time Graphs Lets say that you selected (t,d) as (5,50) and (0,0). Your ∆d = (50-0) and your ∆t = (5-0), hence; Lets say that you selected (t,d) as (5,50) and (0,0). Your ∆d = (50-0) and your ∆t = (5-0), hence; V avg = ∆d = (50 – 0) = 50 V avg = ∆d = (50 – 0) = 50 ∆t (5-0) 5 ∆t (5-0) 5 V avg = 10 m/s V avg = 10 m/s Does it matter which Does it matter which point you choose? point you choose?

Position-Time Graphs Lets say that you selected (t,d) as (4,40) and (3,30). Your ∆d = (40-30) and your ∆t = (4-3), hence; Lets say that you selected (t,d) as (4,40) and (3,30). Your ∆d = (40-30) and your ∆t = (4-3), hence; V avg = ∆d = (40 – 30) = 10 V avg = ∆d = (40 – 30) = 10 ∆t (4-3) 1 ∆t (4-3) 1 V avg = 10 m/s V avg = 10 m/s It does not matter. It does not matter.

Position-Time Graphs Never hesitate to draw on the graph that is provided to you in the following manner. Never hesitate to draw on the graph that is provided to you in the following manner.

Position-Time Graphs Okay; lets describe qualitatively what is happening in each of these situations. Okay; lets describe qualitatively what is happening in each of these situations.

Position-Time Graphs Possible solution for practice A: An object begins to move away from its initial position at an increasingly faster rate. (Very scientific) Possible solution for practice A: An object begins to move away from its initial position at an increasingly faster rate. (Very scientific) Possible solution for practice A: Brittany was originally walking away from a dog, faster at every step, but then realize the dog was angry so she picked up the pace. Later she realized that she was running away. (More realistic) Possible solution for practice A: Brittany was originally walking away from a dog, faster at every step, but then realize the dog was angry so she picked up the pace. Later she realized that she was running away. (More realistic)

Position-Time Graphs Does that make sense? Does that make sense?

Position-Time Graphs Possible solution for practice B: A distant object is being pulled towards another a point, at an increasingly faster rate. (Very scientific) Possible solution for practice B: A distant object is being pulled towards another a point, at an increasingly faster rate. (Very scientific) Possible solution for practice B: Chris, far from home, gets into a cold car. At first it travels very slowly, but over time it warms up and therefore its velocity increases over time. (More realistic) Possible solution for practice B: Chris, far from home, gets into a cold car. At first it travels very slowly, but over time it warms up and therefore its velocity increases over time. (More realistic)

Position-Time Graphs Does that make sense? Does that make sense?

Position-Time Graphs Another way we can look at motion is to use ticker tapes. Imagine a car with an oil leak! If it leaks a drop of oil every second, we can establish relatively how fast the car is going? Another way we can look at motion is to use ticker tapes. Imagine a car with an oil leak! If it leaks a drop of oil every second, we can establish relatively how fast the car is going?

Position-Time Graphs In your opinion, which car is going faster? In your opinion, which car is going faster?

Position-Time Graphs In your opinion, which car is going faster? In your opinion, which car is going faster? The first line: the displacement is greatest. You will notice that the displacements are all the same, so the velocity of the car is constant. The first line: the displacement is greatest. You will notice that the displacements are all the same, so the velocity of the car is constant.

Position-Time Graphs In your opinion, which car is going faster? In your opinion, which car is going faster? The second line: the displacements are small but increasing. The car is increasing its velocity over time. Hence the car is accelerating. The second line: the displacements are small but increasing. The car is increasing its velocity over time. Hence the car is accelerating.

Position-Time Graphs In your opinion, which car is going faster? In your opinion, which car is going faster? The third line: the displacements are very small at first but become very large. The car was probably at rest and then accelerated to a large velocity in a short period of time. The third line: the displacements are very small at first but become very large. The car was probably at rest and then accelerated to a large velocity in a short period of time.

Position-Time Graphs Task 1: Look at the graph below and determine the v average in each case and then write a short story to explain the events. Be creative. Task 1: Look at the graph below and determine the v average in each case and then write a short story to explain the events. Be creative.

Position-Time Graphs Task 2: Look at the graph below and determine the v average in each case and then write a short story to explain the events. Be creative. Task 2: Look at the graph below and determine the v average in each case and then write a short story to explain the events. Be creative.

Position-Time Graphs Lets do a test: Lets do a test: =3&filename=Kinematics_PositionTimeGraphs.xml =3&filename=Kinematics_PositionTimeGraphs.xml =3&filename=Kinematics_PositionTimeGraphs.xml =3&filename=Kinematics_PositionTimeGraphs.xml