Motion Diagrams

One of the tools that can be used to help visualize and understand motion is the motiondiagram. There is no elaborate art involved here; it is simply a sketch. The moving object can be represented by a dot. The motion diagram shows the position of the object as it moves. There is an equal amount of time (∆t) between each successive position of the object in the diagram below.

You can think of it as a stroboscopic picture. (A stroboscope is a light source that flashes on and off periodically.) Imagine that the object is in the dark, except for the strobe. Every time the light flashes on, you can see the object’s position. The ∆t between flashes is always the same, and the path of the motion will be recorded.

This represents a motion diagram for a car moving to the right. The diagram starts at t = 0 s, when the car is at position 3 m. It continues at regular time intervals, until the car is at position 11 m.

In this Exploration you will begin to study the concept of velocity, using a motion diagram that you create with the tape timer. The object you will set in motion for study will be yourself!

Equipment

Tape Timer
Ruler

To begin to investigate motion in more detail, we will start with a few questions.

1. In the motion diagram above, assume ∆t between dots equals one second. How many seconds did it take the car to travel between from the 3-meter position to the 11 meter position? ______

2. If an object moves at a constant speed, how do the distances between dots in a motion diagram compare? Is the car above traveling at a constant speed? How do you know?

3. If an object speeds up, how do the distances between dots in a motion diagram compare?

4. If an object slows down, how do the distances between dots in a motion diagram compare?

5. Drawing sketches showing the motion diagrams for an object traveling in the three different manners may be helpful. Draw a motion diagram of 4 dots for each of the following. Number your dots (0,1,2,3) to show where t = 0 s, and which direction the object is moving

Constant speed / Speeding up / Slowing Down

Tape Timer Activity 1 – Constant speed

Have one person in the group hold the timer steady and set the timing rate to 10 Hz. This means that ∆t between each hammer strikeequals 0.1 second.

6. Here’s a trick question: If we start the clock (t = 0s) when the first dot is made, how many dots will you need to show one second? Hint: the answer is not 10! ______.

Tear off a length of paper tape about 2 feet long and thread one end through the timer. Hold one end of the tape and start walking away from the timer while holding the tape horizontal. Try to hold your speed as constant as possible. Just after you start walking, your partner should turn on the timer. Continue walking until the tape is completely through the timer. If the dots are on top of each other, you went too slow. Observe the placement of the dots on the tape. You have created a motion diagram. The distance between each dot and the previous one represents an equal time interval. If you walked at a constant speed, how does the distance interval between the dots compare? ______

7. Using a ruler, choose a one-second intervalwhere you were moving with fairly constant speed. Number the dots, starting with “0”. Calculate your averagespeed in centimeters per second:

______cm/s.

8. Explain how you calculated the average speed:

9. Is the average speed you calculated a good approximation of your speed at the beginning of the interval? At the end of the interval? Why or why not?

Tape Timer Activity 2 – Changing speed

Repeat the procedure, only this time speed up as you move away from the motion detector. The time interval between each dot is still equal.

10. How does each distance interval between the dots compare? Record your observations:

11. Choose a one-second interval, number your dots starting with 0, and calculate the average speed as you did before:

vavg: ______cm/s

12. This average speed is no longer a good approximation of your speed near the beginning or the end of the interval. Devise a method to approximate your “instantaneous speed” at dot 1 and at dot 9 (recall your first dot is 0).

11: speed at dot 1______cm/s

12. speed at dot 9 ______cm/s

13. Ranking Task: Motion Diagrams - Average Speed

The figures below represent motion diagrams for objects moving from left to right along the x-axis. The time interval between each dot is the same for each diagram. Displacements are given in meters.

Rank these diagrams from least to greatest average speed in the first three time intervals. If two or more diagrams indicate the same average speed, give them the same ranking.

Least 1______2______3______4______5______6______Greatest

Explain the reasoning for your ranking. ______

Ranking Tasks are adapted from A Selection of Ranking Tasks, T.L. O’Kuma, D.P. Maloney and C.J. Hieggelke.

Exploration 1: Drawing Motion Diagrams

Draw a motion diagram for each motion described below.

Six to ten dots are appropriate for most motion diagrams.
Number the positions in order, with “0” being the starting dot. This allows us to distinguish between speeding up to the right and slowing down to the left
Be neat and accurate.

1. A jogger is running at a constant speed to the left, when she starts to tire and slows down to a stop (don’t forget to number the positions in the correct direction).

2. Bob throws a ball into the air and then catches it when it comes down (show up motion going up and down motion going down in 2 separate columns, vertically down the page). up down

3. An elevator starts from rest at the 100th floor of the Empire State Building and speeds up to a constant speed as it descends, with no stops, until it slows down to a stop at the ground floor (Draw this one vertically as well, since the motion is vertical).

4. A car starts from rest and speeds up until it runs out of fuel, then it coasts at constant speed (I’m ignoring the friction of the road here, something you get to do in physics class, but not in real life!)

5. A rocket starts from rest and speeds up vertically until it runs out of fuel. But even ignoring all friction, the rocket slows down when it runs out of fuel, because the Newton’s Law of Gravity can never be broken.

6. A boy lands on a trampoline. Draw a motion diagram for the entire time he is touching the trampoline; from the time he first touches the trampoline, until the time the trampoline launches him back into the air (two columns as well).

downup

Exploration 2: 1-D Vectors

Do agraphical vector subtraction for the initial and final quantities listed below. These are vectors that represent position (x,y), displacement(Δx, Δy), velocity (v) or change in velocity (Δv). Remember to put the tails of the initial and final quantities together, at an arbitrary origin (you don’t have to label origin). Make sure you label the delta quantity (Δx, Δy, Δv) or its wrong! Use the scale below for scale:

Example:

x0 = -2, x = 4: Find the displacement (Δx):

Δx

(Note that Δx = x – x0 = 4 – (-2) = 6, and the displacement vector above is 6 units long, confirming that it’s correct).

1. v0 =7, v =3,2. y0 = 4, y = -2,

Find change in velocity ΔvFind displacement Δy

3. x0 = -4, x = -5,4. v0 = -10, v = -8,

Find displacement ΔxFind change in velocity Δv

5. y0 = 1, y = -7,6. x0 = -2, x =3,

Find displacement ΔyFind displacement Δx

7. v0=7, v =10,8. x0= -4, x = -6

Find change in velocity ΔvFind displacement Δx

9. y0 = 2, y = -2, 10. v0 =7, v =1,

Find displacement ΔyFind change in velocity Δv

11. Which of the following vectors are always in the direction in which the object is moving?

a. positionc. velocity

b. displacementd. change in velocity

12. For each of the following :

Draw an appropriate motion diagram with 3 dots (and therefore 2 velocity vectors).
Draw the acceleration vector above the velocity vectors. Assume a constant acceleration.

a Speeding up in a positive direction:

b. Speeding up in a negative direction:

c. Slowing down in a positive direction:

d. Slowing down in a negative direction:

13. For each of the following motion diagrams:

A. Draw the initial and final velocity vectors. Label them v0 and v.

B. Draw and label the acceleration vector, a, in red.

D. Indicate whether the object is speeding up, slowing down or moving at constant speed. Write your answer besides the diagram.

14. Each of the following describes the observed motion of an object moving with constant acceleration. This means the velocity changes by the same amount each second. The object may change direction (slow down to a stop and speed up the other way). In each case determine:

1) the sign and magnitude of the acceleration vector,

2) what the object is doing (speeding up, slowing down, or changing direction).

a. , ,

b. , ,

c. , ,

d. , ,

15. If the acceleration vector is negative, is the object speeding up or slowing down? If you can’t tell, state that explicitly and state what additional information you must have:

Exploration 3: Kinematic Equations and Freefall, the Movie

Objective: To see where 2 of the kinematic equations come from by analyzing objects in free fall.

Equipment: Computer, mouse (if available)

Log on to Logger Pro and when the default screen comes up go to the “Insert” Menu and choose: Movie, Sample Movies folder, Ball Toss Folder, Ball Toss.

Watch the movie a time or two. The plot is fairly simple. At exactly 1.7 seconds into the movie, the ball leaves the prof’s hands, moving at an initial velocity of just under 3 m/s. It rises about a half a meter in the air and then starts to fall. From the time the ball is released into the air, until it is caught, the only influence on its motion is the force due to gravity. It is in freefall.

Part 1 - Predictions of what graphs for velocity and position will look like from the instant the ball leaves the professor’s hands until the instant he catches it:

Indicate the “turnaround point” on both graphs with an arrow. When your group agrees upon a prediction, show me!

Part 2 – Analysis; Graphs and equations for position and velocity

In this experiment, you act as the motion sensor, and your “sampling rate” is equal to the number of frames per second of the movie.

You can undo your actions by going to the Edit Menu.

Right mouse click to move movie/graph from back to front.

1. Enable video analysis by clicking the bottom rh button (3 red dots and an arrow). This gives you a column of icons along the rh side of the movie.

2. This step synchronizes the scale of the film with the scale of the graph. Set scale by clicking the 4th button down on upper right corner (yellow picket fence icon). Carefully “select” the meter stick on the table. You should get a green line and a pop-up dialog box. Pick the default value of 1 meter.

3. Use the Next Frame button (2 arrows) to find the frame where the basketball just leaves the professor’s hand. This occurs at 1.700 s (time is displayed in upper rh corner).

4.Click the Add Point button (second from the top – red dot in crosshairs). Carefully center the crosshairs on the top of the basketball.. Click once to add the data point to the movie and the graph. If you goof up, go to Undo.

5. Make the trail of blue dots on the movie go away by using the Toggle Trail button (6th from top).

6. Repeat adding points until you determine the place where the professor catches the ball. It’s a pretty delicate operation up at the top of the trajectory where the ball is moving very slowly. Try to be precise, and check the data columns for obvious bad data points.

7. Choose an origin for position. Step backwards through the film until back at your first point (1.700 s). Set the position (vertical axis) origin to your first point by clicking the Set Origin button (third from the top) and clicking on the data point on the movie. A yellow set of axes will appear on the movie, and the blue data point on the graph should be at y=0 (check on the graph, as well as the film). The 2nd button from the bottom is the “show origin” button. Use it to lose the yellow axes, if they annoy you.

8. Hide X (red) data by clicking on the Y symbol on the vertical axis and choosing it from the list.

9. Force the time(horizontal axis) origin to zero. We want 1.7 seconds to become 0 seconds. To achieve this, make a new calculated column under the Data menu. Change the appropriate categories as follows:

Name: Tosstime, Short name: t, Units: s

Equation: Click on Variables (Columns) and choose “Time”. Keep the quotes. Complete the equation by subtracting 1.7 (“Time”-1.7). Note minus sign is outside the quotes.

10. Check the column data to make sure it worked. Replace the time axis with tosstime, by clicking on the horizontal-axis label. Now t=0 s is when the ball was released.

Velocity vs. time graph. – Choose Y-velocity for the vertical axis to bring up a velocity graph of the data.

A. How does it compare to your prediction?

B. Perform a linear fit (R= ) on the data. 2. Record the equation of the best-fit curve and print the graph (put your initials in the footer):

This is one of the kinematic equations. It describes the velocity of an object with constant acceleration, which is the case for an object in free fall. Write it down below in symbolic form:

______

C. Using your text as a reference, explain what each term in this equation actually represents for the ball. Don’t use terms like slope, or y-intercept. Instead, explain what they tell you about where the ball is, or what the ball is doing. Also what is the variable used in your text for this value?

D. Are the values for m and b reasonable for this motion? Explain.

Position vs. time graph (choose Y for the y axis)

A. How does it compare to your prediction?

B. Find the equation that best fits this curve. You probably realize that the linear fit isn’t appropriate; we’ll need another equation type. Use the curve-fit icon [f(x) =]. Choose “quadratic” and then Try Fit.

Record the equation of the best-fit curve and print the graph:

______

This is one of the kinematic equations. It describes the position of an object with constant acceleration, which is the case for an object in free fall. Write the appropriate equation below:

______

A -

B -

C - This term (it should be pretty small in your equation) represents initial position y0. It is not part of the equation in your text, because the text assumes the initial position is zero. I will not always make that assumption!

D. Are the values for A and B reasonable for this motion? Do they agree with the previous kinematic equation?

Questions

1. In both equations there are terms that represent the acceleration of the object. Record the value that you obtained for acceleration (include sign):

______m/s/s.

2. According to your results, did the sign (i.e. direction) of the position (Y) change when the direction of the ball’s motion changed from up to down? Explain why it did or did not.

3. Did the sign (i.e. direction) of the velocity change when the direction of the ball’s motion changed?Explain why it did or did not.

3. Did the sign (i.e. direction) of the acceleration change when the direction of the ball’s motion changed? Explain why it did or did not.