At the point where a speeding driver is caught by a cop, the cop comes up to the speeder and says, "You were going 60 miles an hour!" The driver says, "That's impossible, I was traveling for only 7 minutes. It is ridiculous how can I go 60 miles an hour when I wasn't going an hour?

Getting Position-Time Data from an IP Simulation

When you used the motion detector, the computer took care of all the length and time measurements needed to track motion automatically. In order to understand more about how the motion software actually translates measurements into one dimensional velocities and accelerations it is helpful to make your own length and time measurements for a cart system.

Consider the type of uniformly accelerating cart motion that you recently studied. Suppose that instead of a motion detector you have a video camera off to one side so you can film the location of the cart 30 times each second. (This is the rate at which a standard video camera records frames.) By displaying frames at regular time intervals it is possible to view the position of the cart on each frame as shown below.

A scale diagram of the position of an accelerating cart at 8 equally spaced time intervals. The cart actually moved a distance of just less than 1 meter. Every 6th frame was displayed in the cart movie, so that 5 frames were recorded each second. At each time the center of the cart is located by the number 1.

We will simulate the motion of a cart in this program. The simulation allows us to run experiments in slow motion much like a video camera in slow motion. In addition, the program can display data and graphs.

(a) Run the Program Interactive Physics. Open the file 04Cart. Note the meters on the screen measure position and time. You need to use the Run/Stop/Reset commands on the menu and the tape controls at the bottom of the screen. Do not save the file.

(b) Run the simulation once; then reset it and use the tape control to step one frame at a time in 0.20 second intervals. Record the cart's distance from the origin (i.e. its position) in cm at each of the times 0.00 s, 0.20 s, etc. and fill in column D in Table 1 below.

TABLE 1

Calculating Average Velocity at a Series of Points as an estimate of the Actual Velocity at those Points

Calculation of Average Acceleration at a series of points as an approximation to the actual value of acceleration at those points

Using the same method as for velocity, fill in Column F of the Table

Finding Acceleration from Slope of v-t graph

(a) Create a graph of velocity as a function of time and print it.

Note that in doing this we are assuming that the average velocities we calculated in the table before are good approximations to the instantaneous velocities at the points in question.

(b) Find the slope of the line that fits smoothly through most of the data points. The slope represents the acceleration of the cart. Report its value.

Note: To do this you should use Excel to put an equation and obtain the slope from the equation

(c) How does the value for acceleration which you determined from the slope compare to those you determined earlier from the average accelerations at the mid-point of each time interval between average velocity values. In other words, how does the slope compare with the averages reported in column 5 of the Table ?

1D Kinematic Equations for Constant Acceleration

x = position along the x axis (which can vary with time)

v = instantaneous velocity along the x-axis (which can also vary with time)

a = constant acceleration along the x-axis (It does not vary in time because we have chosen to consider only those cases for which a is constant.)

Beware: The kinematic equations ONLY APPLY WHEN AN OBJECT UNDERGOES CONSTANT ACCELERATION.

The Fundamental Kinematic Equation: [1]

This equation indicates that a graph showing the position as a function of time of any motion with constant acceleration is a parabola of some sort.

Note: All other kinematic equations can be obtained from this one and the definitions of instantaneous velocity and acceleration.

Kinematic Equation #2: [2]

Formal Problem Solving

You should get plenty of practice using these equations for constant acceleration.

TESTING OUT THE KINEMATIC EQUATIONS FOR CONSTANT MOTION TO SEE IF THEY WORK

This time we will not use the IP data. Instead we will use data from the airtrack that we saved before

You should use the Excel Modeling Worksheet and transfer at least fifty position data points from your past experiment file containing motion detector data for an accelerated cart. Paste them into the Position column in the modeling worksheet.

(a) Where does the data you plan to use for distance and time come from? Describe the motion.

(b) Examine your data. What are the values of vo and xo corresponding to the motion you tracked? Enter them into the proper places in the worksheet. The equation of theory column uses these values to calculate the mathematical model for the data.

(c) Theoretical values for position are calculated in the theory column in the modeling worksheet. Adjust the value of "a" until the theoretical curve passes very near or through most of the experimental data points. You may then have to adjust vo and xo slightly to perfect your fit. When you are satisfied with your fit print (with your name in the header) and attach the printout.

Note that if it is possible to get a good fit then the equation must be of the form we are assuming. Otherwise NO combination of numbers would work.

(d) Write down the equation that describes the motion you studied in the form x = f(t)

ACCELERATION DUE TO GRAVITY

Determine g from the dots. Use every dot and be careful to include the effects of any missing dots. You can find "v" for each pair of dots using the definition:

Record the data on the tape. Fasten the tape to your report. Use Excel to plot a graph of v as a function of t; put in a trendline with its equation, use the slope to find g. Note that you cannot use g = 2s / t^2 because the speed at the first dot is not zero.

USE OF CALCULUS IN MOTION (Optional)

Instantaneous Velocity and the Slope of an x vs. t Graph

What happens if we want to know the velocity of an object at a single instant? That is, we'd like to estimate the velocity of an object during a time interval which is too small to measure directly. Since velocity is a measure of the change in position over time, it is possible to use techniques developed in calculus to estimate how a continuously varying function of position vs. time changes during a very short time interval. Let's start by considering how we might determine the slope of a continuous function and proceed from there.

(a) In Figure above what is the equation for the average slope of the curve at the highlighted point in terms of x1, x2, t1, and t2?

(b) How is the value of the slope related to the average velocity in the time interval [t1, t2] ?

(c) Since the rate of change of position is increasing as time goes on (so that the position "curve" is not linear), how can you calculate a more accurate value of the slope?

(d) How would you find the "exact" value of the slope at the point at a time of interest?

(e) Notice that in our position vs. time graph we are interested in how x changes with t. Thus, we would use the notation x(t) to indicate that x is a function of t.

(f) How might the instantaneous value of velocity at the highlighted point be related to the derivative of x with respect to t at that same point? If you can't follow this retrieve one of you distance time graphs (or make a new one) made with the motion detector. With the graph on the screen find and click on tangent under the analysis menu. Observe how the tangent changes as you move along the curve. THIS TANGENT IS THE DERIVATIVE.

(g) Suppose that x(t) = t2 + 1 , where x is in centimeters and t is in seconds. What is the derivative of this function with respect to time? What is its instantaneous velocity at a time t? (You have the equation of the tangent at any point t)

(h) What is the instantaneous velocity, v, in cm/s at t = 1 s? At t = 2 s?

Instantaneous velocity is defined as the time derivative of the function which describes how position changes in time. Similarly, instantaneous acceleration is defined as the time derivative of the function which describes how the instantaneous velocity varies in time. This for us is just a better way of describing the slopes of the graphs at an instant in time. Thus, for an object moving in one dimension

and

Note: The triple bar symbol () is stronger than an equality. It means "defined as".

In the special case in which the variation of a position or velocity with time can be represented by a power of time (btn), the derivative can be taken quite easily. The instantaneous velocity is given by

For example, if x = 5t3 then

The acceleration can be found by taking the time derivative of the velocity function, vx.

Determining v and a by Differentiation

(a) Suppose x= 4t^2 m. Find a general expression for vx as a function of time. What is v at t =0 s? At t = 2 s? (Don't forget units!)

(b)Use the general expression for vx you found in part a to find a general expression for the x-component of acceleration, ax, as a function of time. What is the value of a at t =0 s? At t = 2 s? Is the acceleration constant or does it change in time?

(c) Suppose x= 3t^4. Find a general expression for vx as a function of time. What is the value of vx at t =0 s? At t = 2 s?

(d) Use the general expression for vx you found in part c to find a general expression for the x-component of acceleration, ax, as a function of time. What is the value of ax at t =0 s? At t = 1 s? Is the acceleration constant or does it change in time?

1. If the derivative of the sum of two functions is equal to the sum of the derivatives of the two functions, what is the instantaneous velocity of an object whose position is given by x = 4t^2+3t^4. Hint: Note that this is the sum of the functions in parts (a) and c)

VECTORS BASICS

Most physical quantities may be classified as scalars or vectors.

One of the most common ways to indicate that a quantity is a vector is to represent it as a letter with an arrow over it. Thus, in this guide symbols such as , , , and all represent quantities which have magnitude and direction while r, v, a, and F represent only the magnitudes of those vectors. Note: You should always place an arrow over vector quantities, or else put a stroke under it.

NEGATIVE OF A VECTOR

The negative of a vector is a vector of same length but opposite direction.

ADDITION OF VECTORS

Adding vectors does not mean adding magnitudes.

The arrows have to be added head - to - tail as shown in class.

To calculate vector sums we may use:

• scale drawings, rulers and protractors
• Pythagoras' Theorem
• Sine rule,

SUBTRACTING A VECTOR

To subtract a vector add its opposite.

Most common applications of vector subtraction involve displacements and velocities.

Example: to find average acceleration we need to subtract initial velocity from final velocity then divide by time

Relative motion involves subtracting velocities.

ALGEBRAIC OPERATIONS

A two-dimensional vector can be represented as an ordered pair of numbers(COMPONENTS).

Negative of vector: take negative of components

Add Vectors: add components.

MULTIPLYING VECTORS: DOT PRODUCT

The dot product of two vectors is a scalar of magnitude equal to the product of the vectors and the cosine of the angle between them.

MULTIPLYING VECTORS: CROSS PRODUCT

This is a vector with direction perpendicular to each vector and value equal to their product times sine of angle between them. The direction is given by the right hand rule.

VECTOR COMPONENTS

There are some alternate ways to represent vectors. For example, the diagram below illustrates the representation of vectors through unit vector notation.

rmeas = ___________ xmeas = ___________ymeas = ___________

Computer Simulation:Vector Magnitude and Direction (Components)

To understand vectors more you can play around with this program: Virtual Labs - Vectors - Vector Components

In it you can change one part of a vector and see what effect that has on the other parts.

• Then answer these questions:
• Circle the correct answers to the following multiple-choice questions
• 1. A vector making an angle of 45 degrees with the x-axis has and components of equal magnitude.
• True.
• False.
• 2. A vector makes an angle of 37 degrees with the x-axis. The magnitude of the x-component is
• smaller than its y-component.
• greater than its y-component.
• equal to its y-component.
• 3.A vector is given by its components:Ax=2 cm, Ay = -4 cm. The magnitude of the vector is AND the angle from the x-axis is
• 2.0 cm.
• 4.47 cm.
• 6 cm.
• 27°
• 63°
• -27°
• -63°

TARZAN

• Start up the Virtual experiment Vectors-Relative Motion, or if it is already running, restore the default conditions by clicking the Defaults button (the button with the yellow arrow) or choose from the menu.
• Run the simulation. How does the swimmer move?
• Knowing the two velocities can you add them to find the direction and magnitude of the resulting velocity? Show your calculation of the magnitude and direction below.
• Double the swimmers velocity. How did this affect the angle? How did this affect the time to cross?
• Restore the default conditions, now double the river velocity. How did this affect the angle? How did this affect the time to cross?
• Restore the default conditions, then change the swimmer velocity to 2 m/s. By drawing a triangle with the swimmers velocity as the hypotenuse, the river velocity as one leg, the result as the other leg, calculate the angle the swimmer must head to arrive straight across the river. Show the calculatioN. Then set the angle and run the simulation. If you calculated correctly, the swimmer should go straight across.
• For the angle you found in part 6, which distances and velocities can you use to calculate the time needed to cross the river by t = d/v? There is more than one answer, explain.
• Experiment with other angles, speeds etc. until you are confident that given any three (swimmers velocity, river velocity, distance, time) you could calculate the others and the total velocity for any case that results in a right triangle for the vectors.