23.Mar.15: Trajectories

Purpose: We will set up a scenario that resembles a ball being thrown off a cliff and predict where it lands.

Materials:

• Carbon paper
• Ring stand
• Clamp
• paper
• Steel ball
• meter stick
• Angle Degree measurement device
• Wooden board (Part 2)

You will set it up like this however, in the first part you will not have the wooden board there. 

Procedure: 

1. Make sure you mark where you let go of the ball because you are going to keep using that distance. Watch approximately where it lands on the floor and put the carbon paper over a white sheet of paper there. After 3-4 trials measure where it lands in the x direction and in the y direction. 

OPTIONAL: use a degree measurement to make sure your meter stick is straight in x and y direction. 

2. From this calculate the speed of the ball using the trajectory equations. Make sure to separate the x and y component because they have different velocities, accelerations, and height. Only their times are dependent on each other. (This is JUST like homework problems except we actually came up with the numbers from our experiment) :) 

our speed turned out to be 1.41 m/s

First we used the y motion to find time using x=V(initial)t + 1/2(a)(t)^2

where v(initial) is zero. After we found time we used the same equation but with the x side now to find the v(initial) which equals v(final)

3. Now add the wood cardboard like the picture shown above. Drop it a couple times to approximate where it will land and tape carbon paper to the board. You will then have to find the angle and using calculations determine your value of d (distance). This is where the most trouble came out of our lab because different angle measurement tools gave us different angles. We decided to just use the one that made most sense to our data; however, there is a big uncertainty because the measurements were about 10 degrees apart. 

our calculation for distance

4. Then calculate the change in distance. We started off with the 1/2*gravity*time^2 because Vinitial is zero. and set that equal to distance * sin theta which is the length of the board. We also found that x = vo * t which equals to the distance * cos theta which is the length of the floor. SO we solved for time in each equation and set those equal to each other because time is dependent of each other throughout. We wanted to solve for distance. After solving for distance we calculated the uncertainty.

5. We measured our actual value of x, y, and theta from our experiment. The dx, dy, and dtheta came from uncertainty that we could have been off by by our measurements. In other words it is the +/- of our measurements. You'll see that all of this was listed on the left hand side of our board.So we began with the distance formula that we used to solve. We took partial derivative to each variable (x, y, and theta) which gave us our dd equation in the orange. After coming up with our change in distance formula we plugged in the numbers which is in the green at the bottom of the board. Our distance has an uncertainty of +/- .0159.

Conclusion/Errors:

The biggest error that our lab made was the angle. We measured it with three difference devices and got three (almost totally) different angles. We decided to go with the angle that gave us the best result which was 49.2 but that could have been the wrong angle. Another error was the measurement of the ball. The ball landed some-what in the same spot but it was scattered. So we took the middle one and +/- those few meters for uncertainty.