Calculate the slope of position vs. time graph to calculate the velocity. Determine position from the equation x(t) = v0 t + x0 .: Expert Solutions for Success

Position and Velocity

Objectives:  Students will learn how position relates to constant velocity by predicting positions and velocities of given position vs. time and velocity vs. time graphs.  Students will also calculate the slope of position vs. time graph to calculate the velocity.  Finally, students will determine position from the equation x(t) = v₀ t + x₀ .

Background

Any measurement of position, distance, or speed must be made with respect to a reference frame.  For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.  For today’s simulation we will use the cartesian coordinate plane, xy-axis, to orient our frame of reference.

Distance vs. Displacement

Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.  Displacement depends on direction and thus, is a vector quantity; it can be a negative, or positive value that indicates its direction and magnitude.

The displacement is written: ∆x = x₂ − x₁

Distance traveled (dashed line) is measured along the actual path.   Distance is a length, and is thus a scalar quantity.  Distance can be any measure x ≥ 0 units.

Average Speed vs. Velocity

Speed is how far an object travels in a given time interval

Velocity is speed and includes directional information.  

For constant velocity, non-accelerated motion, we can use the equation v = ∆x/∆t  to solve for either position, or velocity, or time, if we know the other quantities .

  Pre-Lab Questions

1. Consider a deer that runs from point A to point B. The distance the deer runs can be greater than the magnitude of its displacement, but the magnitude of the displacement can never be greater than the distance it runs.

a) True

b) False

 2. Consider a car that travels between points A and B. The car's average speed can be greater than the magnitude of its average velocity, but the magnitude of its average velocity can never be greater than its average speed.

a) True

b) False

 3. Complete the partially filled in table below

 Procedures

Getting Started

1. Go to http://physics.bu.edu/~duffy/HTML5/1Dmotion_graph_matching.html We will use this Physics Simulation to practice with non-accelerated motion in 1D.  Your screen should look like the image below:

2. Please look at the position vs time, velocity vs time graphs, and the motion diagram, which shows the displacement of motion of the red dot.

3. Notice there are Play, Pause, << Step, Step >>, and Reset tabs you will use throughout this simulation to modify your simulation as it plays.  Click Pause, << Step, and Step >> , Reset to change the variables, as necessary.

4. There are 3 sliders that control Initial position (from -50m to +50m), Velocity (from -10m/s to +10m/s), and Acceleration (from -2.0 m/s/s to 2.0 m/s/s).  We will keep the acceleration at zero for all of today’s simulations.

5. Under the 3 sliders there are Match a position graph and Match a velocity graph .  Today we will use Match a position graph.

Graph Matching

6. Review the 5 different Position vs. time and velocity vs. time graphs below.  

7. Predict  the velocity that will determine the position for each of the 5 Position vs. time graphs using the controls on the simulation, explained in the Getting Started steps

8. Go to the simulation Matching Graphs.

9. Click Graph 1, at the bottom of the screen,   Predict how to match this graph by predicting how you should choose the initial position and the velocity values with the sliders at time t = 0 s, 10 s, & 20 s.  Note: Each of these graphs may have one to three different values for position and velocity.

10. Press the Play tab to run the simulation, with your predictions.

11. Did you match the graph perfectly?  If not, change the Initial position and velocity controls until you have matched the graph.  Then record the data from the graphs into the Matching Graphs table below.  You may need to pause the simulation to change the velocity at certain times.  Use << Step and Step >> tabs to slowly increment the motion in time steps of 0.05s.

12. Repeat steps 8 – 10, for Graphs 2 – 5.

13. Calculate distance, displacement, average speed, and average velocity for each of the 5 graphs you matched; enter your results in the table Matching Graphs.

 Questions

14. Is there any instance where the total displacement is greater than total distance? Explain.

15. Is there any instance where the average velocity is greater than the average speed? Explain.

Motion with Constant Velocity:  

16. Now play the simulation, without matching, by selecting the  tab at the bottom of the screen, just below the match graph options.

17. Move the Initial position slider to 0m.

18. Move the Velocity slider, to choose any velocity between 1 and 10 m/s.

19. Set the acceleration slider to 0m/s².

20. Now run the simulation to see how long it will take.

21. Divide the total time of your simulation by 4.

22. Play your simulation again, so that you may now record the position and time for the

23. first quarter of the simulation in the Constant Velocity table below.

24. Continue the simulation until you get to the 2^nd quarter, 3^rd quarter, and the end of your simulation and record those positions and times in the Constant Velocity table.

25. Take a screenshot of your position graph, after simulation runs completely, and enter it here, like this one:

26. Calculate 2 slopes. The first between 0 and the first quarter of your simulation and the second from third quarter to the end of your simulation.

27. Enter your slope calculations in the Constant Velocity table below:

28. Compare the average slope with the velocity, i.e. calculate the percent error between the velocity and your calculated average slope.

29. Now calculate the position, x_2calc, value using your recorded data from the Constant Velocity table and the equation x_2calc(t) = vt + x₀. Enter your results in the Position Data table below:

30. Compare x_2calc with the value x₂ from your data, i.e. calculate the percent error, for your calculated position vs. your measured position.

31. Repeat steps 21 for 2 more positions; choose any 2 positions you like.

Post-Lab Questions

1. The figure shows a graph of the position of a moving object as a function of time. What is the velocity of the object at each of the following times?

a) At t = 1.0 s

b) At t = 2.5 s

c) At t = 4.0 s

d) At t = 5.5 s

e) What is the average velocity of the object from t = 0 s to t = 4.0 s?

f) What is the average velocity of the object from t = 0 s to t = 6.0 s?

2. The graph in the figure shows the position of an object as a function of time. The letters H-L represent particular moments of time.

a) At which moment in time is the speed of the object the greatest?

b) At which moment in time is the speed of the object equal to zero?

 3. If you run a complete loop around an outdoor track of length 400 m in 100 s, find your

a) average velocity and

b) average speed.

4. A polar bear starts at the North Pole.  It travels 1.0 km south, then 1.0 km east, and then returns to its starting point.  This trip takes 0.75 hr.  

a) What was the bear's average speed?  

b) What was the bear's average velocity?

5. Two locomotives 70 kilometers apart are travelling on the same track towards each other , Engine A  moves at  22 kilometers per hour east  and engine B moves at 13 kilometers per hour west.  At the instant both trains begin moving, an annoying mutant fly begins flying from engine A  towards engine B at 33 kilometers per hour . The instant it touches B, it turns around and flies back. It goes on  this way until the two locomotives collide and the mutant fly is finally squashed. 

 So, before its untimely demise, determine  the following:

a. the total distance the fly flew

a. the time it took till  it was eliminated

b. The average velocity of the fly  

Position and Velocity

Introduction

Measurements of position, displacement, and velocity are done referencing a specific frame. For example, when a person is walking in a moving bus, their speed is a few meters per second relative to the bus but when related to the ground, their speed will be much higher because it will be the sum of the speed of the bus and his speed. One of the objectives for the lab was to relate position to constant velocity by predicting the position and velocity of a given position versus time or velocity versus time graphs. Another objective was to calculate the slope of the potion versus the time graph. Also, the equation  was used to calculate the position of the object.

Pre-Lab Answers

1. True. Displacement is taken as the magnitude of the straight line joining the two points A and B from A to B while distance is the total length of the path that the deer traces.

2. True. Speed is a scalar quantity and is given by the total distance traveled divided by the time taken while velocity is given by the displacement of the car over the time taken. Since displacement cannot be greater than the distance traveled, thus, velocity cannot be greater than the speed of the car.

3. Table

 Matching Graphs (filled using data from the Physics Simulation)

Questions

14. No. Displacement is the magnitude of the straight line joining two points and it is always less than or equal to the total distance between the two points.\

15. No. Average velocity is given by the total displacement over time taken while average speed is the total distance over time taken. Because total displacement is never greater than total distance, thus, the magnitude of the average velocity can never be greater than average speed.

Motion with Constant Velocity:  

Percentage error between the calculated average slope and the set velocity is given by

Therefore, the slope of the graph is equal to the velocity of the object.

Now, to fill the table of position data, the formula used was;

 Based on the above analysis, we can conclude that at a constant velocity, the displacement of an object at any time interval is constant. The position of the object moving at a constant velocity can also be calculated using the linear equation. Also, from the results obtained from the simulation, we can conclude that magnitude of displacement of an object between two positions is always less than or equal to the distance traveled by the object between the two positions. The same conclusion applies to velocity and speed.

Post-Lab Questions

2. (a) At the moment J

(b) At the moment I

5. (a) Relative speed between A and B is (22+13)km/hr= 35km /hr

Time of travel before collision:

Total distance travelled by the fly before its demise 

(b) Time it took before getting smashed is 2 hours

(c) Trip 1, 

Distance covered by train A 

Trip 2, 

Distance covered by train B is 2.8km

Trip 3, 

Net displacement = 50.22km-10km+5.17km-…=46.22km

Appendix

Sources

Physics Simulation accessed from http://physics.bu.edu/~duffy/HTML5/1Dmotion_graph_matching.html

Appendix