# PHY 141 Lab 3 - Projectile Motion

The purpose of this lab is to study projectile motion of an object which is launched horizontally and drops a certain height before it hits the ground.

## Equipment

• ramp shaped like a “ski jump” with a horizontal positioning screw
• clip positioned on down ramp for placing the steel ball reproducibly at same position
• steel ball
• photo gate
• interface box (photo gate –> computer)
• computer
• ruler
• sheet of carbon paper
• sheet of white paper
• poor man’s “plumb bob” (string with paper clip)

## Introduction

This experiment presents an opportunity to study motion in two dimensions. We study projectile motion, which can be described as accelerated motion in the vertical direction and motion at uniform velocity in the horizontal direction. An object launched horizontally with a velocity vx and dropping a height h, has the following relation between its horizontal distance traveled and vx :

$x= v_{x}\Large \sqrt{\frac{2h}{g}}$
(1.1)

Derive this result for yourself!

## Determine experimentally the relationship between horizontal distance and velocity

You will establish the proportionality constant between the horizontal distance, x, and the horizontal velocity, vx, by studying the motion of a steel ball launched from a ramp.

• Measure the height, h, the vertical distance from the end of the ramp down to the floor and record it on your worksheet. Use the “poor man’s plumb bob” (string with paper clip attached) to define the point vertically downward from the point where the string touches the pulley tangentially. Assume that h has an absolute error of 2 mm.
• Measure the “effective diameter” deff of the steel ball. The effective diameter is the diameter that is “seen” by the photo gate. The measurement of deff may be accomplished by moving the photo gate from the front to the back of the steel ball on the launching ramp. Make sure your platform with the ramp is solidly clamped to the lab bench when you turn the wheel displacing the photogate. The front and back are indicated by where the photo gate is blocked or unblocked as seen by the LED (light emitting diode) on top of the photo gate. The LED is on when the gate is blocked. You read the displacement on the scale attached to the platform. Record your measurement and estimate the absolute error in your measurement for the diameter. Note that once you have measured your deff, DO NOT change the position of the photo gate because shifting the photo gate will change deff.
• Drop the steel ball from the lowest mark on the ramp and note where it lands on the floor, then repeat for the highest mark on the ramp. Tape the carbon paper with a piece of white paper underneath it to the floor, so that the ball will hit the paper when it is launched from the ramp.
• Test the photo gate: block the photogate beam with your finger and see the red light on the cross bar of the photogate turn on.
• Select the program “Precision Timer” on the desktop if you are not already in the data collection program.
• Select Mode: G - Gate Timing Modes
• Select Timing Mode: A - One Gate
• Select Display Type: N - Normal Display Type
• With these settings each time the ball passes through the photogate the time it takes to do so will be recorded.
• Drop the ball from the lowest mark on the ramp, and the computer will record the time t for which the ball blocks the light beam. You should drop the steel ball several times and record the time and the distance measurement. Make sure you position the steel ball reproducibly at the same location on the ramp on each drop. When you think you have enough data points for a particular height, (3-5 measurements will do), press <Enter> to Exit. You can now call up a table of data by selecting the option T - Display Table of Data. As well as your data points this displays the Mean of these values and , the Standard Deviation and the Standard Deviation of the Mean (SDOM). Record everything in your lab notebook. For the distance data, you should make a reasonable estimate of the average distance the ball travels for each launch height and the error in that average When you are done with looking at the table of data you should choose X - Return to Main Menu and set yourself up to launch the ball from a different height.
• Launch the ball from 4 more heights (i.e from each of the markings on your ramp), for each height making several launches and finding the average and error of the average for both the time the photogate is blocked and the horizontal distance the ball travels.

The horizontal velocity from each height is then calculated using the formula

$\Large v_{x}=\frac{d_{eff}}{t}$

and the error in vx should be calculated using equation E.7 from your error manual.

## Graphical methods

Once you have calculated these values you need to make a plot with x on the vertical axis vs vx on the horizontal axis.

In this lab we will use two approaches to this, plotting both by hand and using the computer.

### Making a graph by hand

When you make a graph by hand you should use graph paper. Draw and clearly label axes and choose a suitable scale for your data. Represent the error in your values using error bars which extend a distance $\sigma$ from your value, where $\sigma$ is the absolute error in the quantity. Remember there may be significant errors in both quantities. In some cases the errors in one quantity will be very small relative to the scale of the graph and you may omit them. You need to determine a line of best fit, and measure it's slope. Then you need to draw “max” and “min” lines which correspond to the lines of maximum and minimum slope which fit within the error bars. The slopes of these lines will allow you estimate the error in the slope. In a case such as the present we know the lines should pass through (0,0), this knowledge allows us to put all three lines through that point, substantially reducing the overall uncertainty in the slope.

### Making a graph using a computer

While it is important to know how to plot data by hand, in practice physicists will normally use a computer to plot data. You can use the plotting tool provided for this. You should decide which value of the slope and it's uncertainty you think is a better representation of the data and use this for your subsequent analysis. The plot you make should be printed out and pasted in to your lab notebook. Unfortunately there is no printer in the lab or in the helproom, but the plotting tool will let you send your graph to your email account to be printed out later.

## Calculating the value of the acceleration due to gravity

We need to calculate the value of the acceleration due to gravity from the slope of our graph. The slope of our graph, which we will refer to as k, is related to g through equation 1.1, i.e. $k=\sqrt{\frac{2h}{g}}$. We can rewrite this equation as

$\Large g=\frac{2h}{k^2}$
(1.2)

Now you need to calculate what the value of g should be from the value of k you obtained from the slope of your graph and the value of h you measured earlier. Be careful, if you measured your value of h in cm, you will want to convert it to m so that your value for g has units of m/s2.

Our final task is to estimate the error in our measured value of g. Both h and k have a certain degree of uncertainty and strictly speaking we should take both into account in our estimation of the error in g, though in practice you may find that one error is far more significant than the other. You can find the relative error in g from the relative errors in h and k2 using equation E.7 from the error manual. The relative error in k2 is obtained from the relative error in k using equation E.8 from the error manual. We can then arrive at an expression for the error in g (which you should derive for yourself):

$\Large \sigma_{g}=g\sqrt{(\frac{2\sigma_{k}}{k})^{2}+(\frac{\sigma_{h}}{h})^{2}}$
(1.3)

Finally, you should check whether your measured value of g is consistent with the accepted value of 9.81 m/s2.

Make some notes about how you could improve or expand the experiment. 