Physics HW 2

115L Lab Two

Using Graphs to Recognize Mathematical Relations:

Displacement, Velocity and Acceleration

1 Introduction

Many of the things that are studied using physics are processes that change over time. A system being studied will have some initial configuration at the beginning of the observation. As time goes on, the system will evolve into a new configuration. One of the most common practices in scientific study is to keep detailed records of a systems configuration through a period of time. This allows the observer to look back at the systems evolution as a function of time. Often, the function will follow a consistent mathematical relation that leads to the equations used to model physical processes and predict the behavior of systems.

Today you will look at a simple system, a cart moving in one dimension along a track. You will look at separate functions for the cart’s position along the track, the rate of change of that position (the velocity), and the rate of change of the velocity or amount that the cart is speeding up or slowing down (the acceleration). Plots of the functions, all functions of time, reveal the simple mathematical relations that each follows. Features of the plots also clearly show the connections between the three functions: position, velocity and acceleration. Theses connections are what led Newton and Liebniz to invent calculus. Study- ing the connections between the graphs of these functions is an illuminating illustration of the basic principles of calculus, and how they appear in the phys- ical world.

2 Making a Graph of Position as a Function of Time

You have a track set up at a low angle, and a low friction cart to roll down the track. At the top of the track there is a motion detector. The motion detector sends an ultrasonic pulse, and then detects the reflection. The time delay be- tween the sent and received pulse allows the detector to calculate the distance to the object that the pulse reflected from. A series of these measurements over time allows the motion of an object to be tracked.

1

• Make sure the detector’s switch is set on the narrow-beam setting.

• Make sure the detector is aimed parallel to the track.

• Open the DataStudio file PS115L.Position1.

• Hold the cart about 20 cm from the detector. Make a note of the starting position:

• Click “start” on the DataStudio interface and release the cart. Try to time your release of the cart as close as possible to the start of the data collection. Click stop just as the cart reaches the bottom of the ramp. Catch the cart as it reaches the bottom!!

• Identify the initial position of the cart. The data table may have several readings near the beginning of the data set with values that do not vary, or vary only slightly. Use the average of these values as the initial position of the cart before the release.

• Identify the last relevant data point. If you timed your “stop” correctly, you will be able to use the data all the way to the end. However, if the cart was caught, or struck the end before the recording was stopped, you will have to truncate your data. If you can’t tell where to stop from the table, you will be able to tell during the plotting.

• Record your data in the first two columns of table 1 on page 19. You can leave the third column empty for now. It will be used later.

On the provided tape mark an initial position. Then measure and mark each consecutive position. Remember, the time interval between every two consecutive points is equal.

What do you notice about your position marks as the sequence progresses?

What does this tell you about the speed of the cart as it makes its way down the ramp?

Making a plot of position as a function of time. Plot each point from the table with position on the y axis and the time on the x axis. In this case, t would be a more appropriate label to use for the x axis. What is the coordinate you’re plotting on the y axis?

The position is measured as distance away from the detector. Since the detector is aimed at an angle straight down the ramp, it gives the distance the cart has traveled at that angle, a combination of vertical and horizontal movement. So the position measured in this experiment is not what are commonly called x (horizontal) or y (vertical), but actually the hypotenuse of a triangle made by a combined motion in two dimensions. Of course, what label you give a quan- tity doesn’t really matter as long as the meaning is understood. Some common letters used to label position in a direction that isn’t necessarily defined as “hor- izontal” or “vertical” are d, r, and s.

What type of function does your plot look like?

Does your plot look like the path the cart took down the ramp?

Of course the cart went straight down the ramp. The plot looks quadratic–like half a parabola–because it is a picture of the cart’s motion in time. Remember the previous lab and the equations that described the ball’s position as a func- tion of time after it was fired from the ballistic pendulum:

x = x0 + v0xt + 1

2 axt

2

y = y0 + v0yt + 1

2 ayt

2

You then determined specific values for each term in the general equation sep- arately for x and y. You can do the same thing for the cart on the ramp. Start with the general form of the equation, use s, its a good letter for position:

s = s0 + v0st + 1

2 ast

2 (1)

Now, s0 is the initial position. It should be whatever value the position sensor read before you released the cart. If your data doesn’t give exactly the same reading for the repeating value before the release, use the average of the values at the beginning of your data set. This small fluctuation is just the uncertainty in the instrument’s reading.

v0s is the initial speed that the cart was moving along the line of the detector. What should this value be?

Since the cart was not moving initially, but then it started moving, there must be a force acting on it–giving it an acceleration. Remember, acceleration is a change in velocity over time. The cart wasn’t moving, then it started moving, then it continued to speed up on its way down the track. You probably already

understand that gravity is responsible for the cart’s motion down the ramp, but its not the full force due to gravity acting here. The cart would obviously reach the bottom faster if it just fell straight down. The amount that it speeds up as it descends depends on the steepness of the track. The expression for the acceleration in this case depends on the angle of the track:

a = g sin θ (2)

Here g is still the gravitational acceleration you used previously, 9.8 m s2

. θ is the angle of the track above horizontal. Later in the course you’ll get a chance to see why this is the acceleration for this situation.

The acceleration is causing the cart to speed up in the same direction that you are measuring as a positive distance. What sign should you give the acceleration term in the equation?

So the expression for the cart’s position, as a function of time, when it is re- leased from rest to roll down the ramp is:

s = s0 + 1

2 g sin θt2 (3)

The cart starts at some distance away from the sensor, recorded in your data, and then accelerates in the positive direction–defined as down the ramp in this case by the sensor. The amount of acceleration depends on the angle of the ramp, θ, and gravity, g.

• Measure θ, the ramp’s angle above the horizontal. Use two differ- ent methods to determine the angle. You have different tools provided, including a protractor, level, carpenter’s square, and plum-bob. Remember that trig functions provide a way to calculate angles from measurements of the sides of a right triangle.

• Take the average of your two measurements, and record that as your θ.

• θ =

You will need the acceleration term, g sin θ, for your calculations. cal- culate g sin θ using your θ and 9.8 m

s2 for g and record the value here:

Use your s0 and θ to complete your position equation. s = s0 +

1 2 g sin θt2

• Open Datastudio file PS115L.Position2.

• Record another run of the cart down the track. Try to start in the same position as your previous run (Recorded on page 2).

• This time Datastudio has produced a graph of the cart’s position as a function of time, rather than just a data table of ordered pairs.

• Compare the new graph on the computer with the graph you pro- duced by hand for the previous run. Pay attention to things like the initial and final positions, and the slope of the curve across the domain.

• List at least two things that are similar on the two graphs.

• List any things that are different, if there are any.

• Propose an explanation of what could have caused any differences you see.

• Open Datastudio file PS115L.Position3 • There is a dialog box where you can enter the values for your ini-

tial position, s0, and your acceleration, g sin θ. Take these values from your equation 2 on page 6.After entering the values, click the “accept” button.

• Hold the cart in the same initial position as the previous runs.

• Click “start” and release the cart. Click “stop” and catch the cart at the bottom.

• Datastudio has produced two curves on the same graph. One is a plot of the carts measured position during its descent, the other is a plot of the position function, s = s0 +

1 2 ast

2, with your values entered for the coefficients.

• Compare your theoretical curve to the one plotted based on the measured points.

• On page 6 you calculated an average of two measurements to determine your θ. What effect would using either of the measurements as θ instead of the average have on your theoretical model?

• Would either value of θ give you a theoretical result closer to the measured curve?

• The model assumes an object moving under the influence of a single force. Give three things that may influence the motion of the cart as it travels down the ramp that are not considered in the model.

3 Velocity as a Function of Time, and It’s Rela- tion to Position

Velocity is the change in position over time. The units for velocity will always be some measure of distance, divided by a measure of time: miles per hour, kilometers per hour, furlongs per fortnight, and the S.I. meters per second are all examples of velocity units. Velocity also includes a direction. If you are driving due north, your velocity in the east direction is zero. In this lab, where you are concerned with motion in one direction — a straight line measured out from the motion detector — velocity can have two directions: Away from the detector (defined as positive), and toward the detector (defined as negative).

If a certain distance is traversed by an object in a certain time, the object’s average velocity can be calculated simply by diving the distance traveled by the time taken to travel the distance. To approximate a measurement of actual velocity, which is an object’s speed and direction of travel at one specific instant of time, you can calculate an average velocity over a very short time interval. The shorter the time interval that you measure a change in distance over, the closer you get to a measurement of the objects actual velocity.

Using the data you collected in table 1 on page 19, calculate the average velocity between each two consecutive position measurements. Do this by dividing the distance traveled from the first measurement to the second by the time interval between the two measurements. Use the calculated velocities to complete table 1.

Create a graph of velocity vs. time using the data in table 1.

• Open Datastudio file PS115L.Velocity1. • Record a run of the cart down the track. • Datastudio has created plots of both the position and velocity

• Choose a point somewhere along the curve of the position graph.

• Use the Datastudio Slope Tool. Positioning the slope tool on your chosen point will give you the graph’s slope at that exact point.

• Find the point on the velocity graph at the corresponding time of your chosen position point.

• The X,Y Tool will give you the coordinates of that point. Com- pare the y-value of the point on the velocity graph to the slope of the position graph at the same time coordinate.

• what do you find?

• Repeat the process for 3 different points along the curve of the position graph.

The slope of a graph is often call “rise over run”. For the position graph, the rise is a measure of distance, meters, and the run is a measure of time, seconds. Rise over run is meters per second, the units of velocity. The slope of a graph is the rate of change of the function represented by the graph. Since velocity is a rate of change (over time) of the position, the slope of a graph of position versus time will be the velocity. This relation of a function to its rate of change is the

derivative defined by calculus. By taking the position as a function of time, a corresponding equation for the velocity as a function of time can be obtained by using the derivative. The position:

s = s0 + v0st = 1

2 ast

2 (4)

Taking the derivative with respect to time gives an expression for the velocity:

vs = v0s + ast (5)

Look at your equation for position, by taking the derivative with respect to time, or by comparing with equation 5, find an expression for velocity based on your measurements.

Compare your expression’s coefficients with those of the curve-fit from the Datastudio graph of the velocity.

4 The Acceleration

The equation for position, equation 1, is based on assumption of a constant force providing an acceleration in the direction the equation applies to. This works well for things falling, or otherwise moving with gravity providing the only force driving the motion. This is true because gravity is so close to being constant in the vicinity of Earth’s surface.

• Knowing that the acceleration should be constant for the cart descending the ramp. what should a graph of the acceleration look like?

• Can you predict what the value of the acceleration is from the information contained in the velocity graph?

Just as velocity is the rate of change of position over time, the acceleration is the rate of change of velocity over time. In terms of a mathematical function, acceleration will be given by the derivative of velocity with respect to time. The same information comes from a graph of a function by looking at the slope.

• Use the slope tool to check the slope of the velocity graph in 4 different places. What do you find?

• What does this tell you about the acceleration?

• Looking at equation 5 for the velocity, can you take the time derivative of it? If you haven’t studied derivatives in calculus yet, can you predict what it should be knowing that acceleration in this case is constant?

• Open Datastudio file PS115L.Acceleration1 and record another run.

• Datastudio has now plotted graphs of position, velocity and acceleration.

• How does the graph of acceleration compare with your predic- tions?

• How does the acceleration graph compare with the slope of the velocity graph?

• What is the slope of the acceleration graph?

• What does this tell you about the rate of change over time of the acceleration?

5 Motion Toward the Detector

The detector is set up to measure the distance in a straight line away from it. You saw that velocity of the cart away from the detector had a positive value, increasing in magnitude as the cart speed up. You chose acceleration to be positive because of the observation that the speed of the cart increased as the cart moved in the positive direction.

• Is it possible to measure a negative position with this device?

• Could velocity be negative?

• What type of motion might cause a negative velocity?

• What would happen to the speed of a cart if it was given a push up the ramp?

• What does this mean for the acceleration?

In the space below, sketch how you think plots of position, velocity and acceleration, all as functions of time, would appear if the cart was started near the bottom of the track and given a slight push up.

Describe in words the motion of a cart given a gentle push up the track from the bottom.

Do your graphs match with your description? Identify at least one feature from each graph that matches your description of the motion.

• Position:

• Velocity:

• Acceleration:

Use the equipment to record a run, giving the cart a slight push up the track toward the detector. Stop the recording when the cart reaches the bottom again.

Make comparisons between your predictions and the actual graphs. List at least one similarity and one difference for each graph.

• Position:

• Velocity:

• Acceleration:

Compare your verbal description with the plots.

• What do you notice about the slope of the velocity graph?

• What happens to the sign of the value of velocity as the cart changes direction?

• Does the acceleration change with the direction of the cart’s mo- tion?

Table 1: Table for graphing position and velocity

time (seconds) position (meters) velocity (meters per second)