\"Projectile Motion and Introduction to Errors\" Range and Error in Range Consider firing cannonballs. Will every single one fall in the same spot? You might be surprised to find that there is a range about which they fall, even if the aim is the same each time. Most of cannonballs will fall in the range from X - 5X to X +6X. We will explore this concept in more detail in this lab. You are going to test the effect of two variables on the range of a projectile: the diameter of the ball, and the height from which the projectile is launched. You will test one variable at a time while keeping the other variable constant. You will be looking for a systematic change (probably an increase or decrease) in the range as you change the value of the test variable. Experimental procedure: The balls will be launched by rolling them from various positions from a curved ramp clamped to the edge of the table. The end of the ramp is clamped to the edge of the table so that the ball is launched horizontally. The horizontal range is measured from a point directly under the end of the ramp to the point where the ball hits the floor. The measurements are made using pieces of white paper taped to the floor covered with carbon paper. The point beneath the end of the ramp is marked using a pointed plumb bob on string hung over the end of the ramp. Where this point touches the floor is used as the starting point of the range. The end point of the range is a circular mark made by the ball as it lands on the carbon paper. Take care to catch the ball on the first bounce to prevent extra and potentially confusing marks. Even if you release a ball from the same point on the ramp, you may not get the same range for each release. There will be a variation in the range. You will need to estimate this variation and use it as your experimental error in the range. A way to do this is to take multiple measurements with the same ball and height and examine how the range varies. 1.Release one ball five times from a height chosen by you. Record that height. Measure the range for each trial by recording the horizontal distance of the mark on the carbon paper from the edge of the table. 2.Use Excel to calculate the average of your range measurements, the standard deviation, and the standard deviation of the mean. The average represents your best estimate of the true range, R, of the ball for that height. The standard deviation of the mean is your estimate of the variation in the range for this height which you will use as 5R. Question 1. Describe how you determine the initial height and range of the ball. Draw a simple diagram and label your height and range. Question 2. For the chosen height what is your measured range and the standard deviation of the mean for this range? Testing the Effect of Launch Height on Range Now you will investigate the effect of launch height on the range of the ball. Use the same ball that you used in the previous section. Launch the ball from at least five different heights (five trials for each height). You may use the data you recorded from the previous section as one of these heights. Measure and record the range for each trial. Calculate the average range, standard deviation, and the standard deviation of the mean for the range recorded for each height. Tabulate your results within Excel, and graph the average range against the height using error bars. Plot the data as a scatter plot the length of the error bar is twice crmean: ne by \"mean ' \"mm A template for plotting your data (with error bars) is provided on BB. Note that the error bars may be very small (even smaller than the data point marker!) if the clustering of your measured range is narrowly distributed. Question 3. How does the range generally change as the launch height is increased? Question 4: Which is bigger, the standard deviation, or standard deviation of the mean? Explain the difference between the two using some pictures and a couple sentences Question 5. Can you conclude that the launch height has an effect on the range? Explain what your evidence is. Since the theoretical relationship you determined between range (R) and height dropped (h) goes like R at \\Ih, then there should be a linear relationship between R2 and h. Plot R2 as a function of h, and fit a linear trend to this line. Question 6. Could you use the linear fit that you made to predict the range for any height? If so, how would you go about doing this? Conclusion: Please write a few sentences for your conclusion that addresses: . What variables did you explore the connection between today? 0 What was the general trend of the relationship between these two variables? 0 Report at least one key quantitative result that you measured. - List any reasons that you can think of for why the theory (R at \\Ih) did not match your experimental measurements perfectly