March 24, 2015 Trajectories Lab

Objective: Apply the knowledge of trajectory motion to predict the location as a ball falls from an inclined board.

The Set-Up:

We set up the above apparatus by putting together two metal v shaped bars. We take one bar and lift it up to a certain angle and stable the two metal bars with tapes. Then we test the apparatus with the ball to make sure the structure is stable.

Data Collection: 

we try five more times with a carbon paper placing at the spot where the ball falls, so that when the ball falls, the carbon paper immediately record down the exact location.

From the table to the dots is the horizontal distance. Since the table has rim, in order to accurately measure the distance, we hang a weight on a string when measuring. We measure the distance of each block dots from the hanging weight. Here is our data.

D1= 65 cm

D2= 64.5 cm

D3= 64.9 cm

D4= 64.8 cm

D5= 64.8 cm

We use 64.8 cm as our horizontal distance, with an uncertainty of +/- 0.2 cm. We also measure the vertical distance, the height of table, to be 94. 7 cm with an uncertainty of +/- 0.1 cm.

Putting the data together, we solve for the horizontal initial velocity. The horizontal velocity is 14.7 cm/s, which is 0.147 m/s.

Now we have the initial velocity of the ball, the big question that we still need to solve is if there is a board connecting to the exit of the structure, such that when the ball exit the structure, it falls on the board, what is the distance from the exit to the location where the ball falls?

Our job is to calculate the location, the impact point, then set up the board to confirm our result.

We measure the angle of the board to be 48.7 degree.

Here is our calculation

The distance we find is 76.1 cm or 0.761 m.

Frontal view of the apparatus with the board

Here is the experimental result.

The scale is in metric system. The experimental result is about 77.5 cm or 0.775 m.

Now we use the data to calculate the uncertainty in this result. To find the uncertainty of the distance, there are a few components to consider. These components are the angle of the board, the gravity, and the initial velocity, which includes the heights of the table and the horizontal distance from the rim of the table to the dots. These components are obtained through measuring, thus, there are some uncertainty involve. We take the partial derivative of each component. The first picture below shows the process of getting the partial derivative of initial velocity. The second picture shows the process of getting the partial derivative of the angle and the initial velocity. Gravity is not taken into account because the value of gravity, 9.81, is accurate. After much derivation, we find the uncertainty value for distance to be 0.775m+/- 0.004m

Finally, we also calculate the percent error. The percent error of this distance is about 1.84%.

Summary:

In this trajectory lab, our goal is to predict the exact location of the ball as it exits the track and falls onto the board. We begin by building the track. We did a few trial runs to see where the ball would land without the board. We mark our result with carbon paper then use it to measure the horizontal distance from the rim of the table. Then, we measure the height of the table to be our vertical distance. By having these few pieces of information, we calculate the initial velocity of the ball.

We set up the board and measure the angle. We use the initial velocity from previous calculation, the angle between the board and the ground, and some knowledge of trigonometry and trajectory, to find the distance from the rim of the table to the impact location. We run the experiment; the experimental result is not far from our calculation!

Finally, we calculate the uncertainty of this distance because some components are obtained through measuring. We found the distance with uncertainty to be 0.775m+/- 0.004m. And the percent error is 1.84%.