March 23, 2013; Lab 5: Trajectories

Purpose:

To use projectile equations to derive an expression for point of impact distance on an incline board   along with calculating the propagated uncertainty.

Experiment:
Part 1
1. For the first part of the lab, we let a ball roll of from the top of an incline for it to have some value of velocity when leaving the table (as shown in the setup below).
2. Then we record the distance it travels by the mark it leaves on carbon paper. Several trials are used for purpose of precision.
2. We measure the height of the table and the average value of the distance the ball travels.
our data: h=.881m; Δx= 0.618m.
3. Using kinematics for projectile, we can use the above data to find initial velocity (the velocity right before the ball leaves the tip of the table) as shown below:

- Initial velocity in y coordinate is zero, hence h= (v-initial)t + (0.5)gt^2 = 0.5gt^2
- Δx=0.618m is an average from several trials.
Part 2:
1. An incline is attached to the table with angle θ with the floor such that when the ball rolls down, it hits the incline a distance "d" from the tip of the the table.
2. Since the setup on the table doesn't change, we can use initial velocity from part 1 and derive an equation that only uses θ and Vo to calculate "d".
3. Using right angle identity, as shown below:

At a measured 44.1 degree, actual "d" should be 0.587m 

3. After we find what "d" value is supposed to be, we proceed to the experiment. By attaching an incline to the end of table, keeping the 44.1 degree as we first measured, let the ball roll down and hit the board that will leave mark on the attached carbon paper, we collect several trials for "d". As shown in the picture above, our experimental value is 0.6032m.

apparatus of part 2 lab

4. Next, we calculate the propagated uncertainty.

5. Since the initial velocity is calculated, we have to replace it with the equation that has measured values in it. Hence Vo= Δx*[g/(2h)]^1/2

6. Through partial derivatives, (taking derivative of "d" with respect to x, h,θ) and these following equations are obtained,

- Notice that dθ has to be converted to radians.

7. Our assumed dθ, dx, and dy get us the above value, although the theoretical d= .587m is not in our range.

Conclusion:
 Since initial velocity is used in all parts of this lab, one of the keys to more accurate data is to keep the setup on the table unchanged. That way, we have the same initial velocity throughout the experiments. In addition, although our theoretical value is not in the range, our closest experimental "d"= .5978m, which is approximately 2% error from what "d" should be. The most reasonable argument here is that we pick our  dθ, dx, and dy to be too small. Nonetheless, we can conclude that the experimental data makes sense with the theoretical data we use projectile equation to find.