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.

Important! You need to print out the 1 page worksheet you find by clicking on this link and bring it with you to your lab session.

If you need the .pdf version of these instructions you can get them here.

• 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)

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 v_x and dropping a height h, has the following relation between its horizontal distance traveled and v_x :

You will establish the proportionality constant between the horizontal distance, x, and the horizontal velocity, v_x, 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” d_eff of the steel ball. The effective diameter is the diameter that is “seen” by the photo gate. The measurement of d_eff 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 on your worksheet and assume the absolute error for the diameter to be 1 mm. Note that once you have measured your d_eff, DO NOT change the position of the photo gate because shifting the photo gate will change d_eff.

• 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.

• Now, connect the photo gate output to the interface box by plugging its cable into the top socket (labeled “DIG/SONIC 1”) of the black interface box (“LabPro”). 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.

• Turn on the computer and check the system; Double click the icon “Exp3_t1_t2”. A window with a spreadsheet on the left (having a “Time, column) comes up. On top is a window “Sensor Confirmation”. It should show:

[If you don’t see the above do the following: Click Experiment→Set Up Sensors→Show All Interfaces→DIG/SONIC1: you can check the photogate by blocking it and seeing “Unblocked’ go to “Blocked”. Click “Close”]

• Click OK

• Click Experiment→Start Collection.

• Drop the ball from the lowest mark on the ramp, and the computer will record the time t₁ when the ball enters the light beam and t₂ when the ball leaves the light beam. The difference t₂-t₁ is the time the effective diameter d_eff traverses the beam of the photogate. You measure the horizontal distance x in the following way: you hang the plumb bob from the end of the ramp and measure the distance between the point where the plumb bob touches the floor, and the mark the steel ball makes on the white paper. (Note: do not shift the position of the paper at all until you are finished with all measurements.) Each time the ball passes through the photo gate, the time pair t₁,t₂ appears on the screen and you record the difference (t₂-t₁) in the table on your worksheet. For each mark on the ramp, you drop the steel ball 3 times and record the difference (t₂-t₁) and the distance measurement. For each mark make sure you position the steel ball reproducibly at the same location on the ramp. (Since there are 5 marks on the ramp, you should have a total of 15 time measurements and 15 distance measurements).

A tool is provided to help you make some calculations from the data in your table. For each mark on the ramp we find the average of the values of t and x using 1.5 from Lab 1 and enter them into Table 2. The error of the average t and x is calculated using expression (1.5b) from Lab 1. The horizontal velocity is then calculated using the formula

$\Large v_{x}=\frac{d_eff}{t_{avg}}$

and the error in v_x is calculated using equation 1.7 from Lab 1.

You don't need to do all these calculations, you just enter your recorded values below and click submit and the calculated values will appear on a new tab. Note that you should enter all distance in meters, not cm.

Copy the values the computer gives you in to the second table in your worksheet. Once you have your values written down you need to make a plot of x_avg on the vertical axis (y1,y2,y3,y4,y5) vs v_x on the horizontal axis (x1,x2,x3,x4,x5) using the plotting tool below. Should you include the point (0,0) in your fit this time? (Ask yourself what the distance traveled in the x direction is if the horizontal velocity is zero).

We now want 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 3.1, i.e. $k=\sqrt{\frac{2h}{g}}$. We can rewrite this equation as

Now you need to alculate 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. You can find the relative error in g from the relative errors in h and k² using equation 1.7 from Lab 1. The relative error in k² is obtained from the relative error in k using equation 1.8 of Lab 1. We can then arrive at an expression for the error in g:

Sometimes when we combine relative errors like this we can neglect one of them if it is much smaller than the other. Below there is a tool that calculates the error in g using both the error in h and k, just k, or just h. Enter your values (including your measured value for g, not the official value) and click submit. The computer will display the three error estimates. Based on these three values can you conclude which is the most significant source of error in the experiment? Can you think of ways you could reduce this error?

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

When you've done all these things you are ready to discuss your results and conclusions with your TAs.