March 15, 2017 Lab 5: Trajectories

Lab 5: Trajectories

Chris Ceron, Amy Chung, John Choi

Purpose

To apply our understanding of projectile motion to predict the impact point of a ball on an inclined board

Materials

• Aluminum "v-channel"
• Marble
• Board
• Ring stand
• Clamp
• Paper
• Carbon Paper

Theory/Introduction

We set our apparatus by connecting two v channel rails to create a ramp. The first rail was set at an incline to provide the marble with velocity. The second rail allowed the ball to launch horizontally.

Part 1: Determine the initial velocity of the ball

Procedure:

In order to record where the marble would hit the floor, we launched the ball and noted where the ball lands. Then we taped a piece of paper on that location, and added carbon paper on top to create marks each time the marble landed. 

We then launched the ball five times from the same place and ensured that the ball would land virtually in the same place each time.

After repeating the process five times, we recorded the height of our apparatus by taping a piece of string with a weight onto the edge of the table (also where the marble launched out of the v-channel) and measuring along the string. The weight on the string allowed for the string to fall straight down and perpendicular to the floor. To record the distance the marble traveled, we set our meter stick to start from the end of the hanging string, and measured to the center of the five landing points. We assigned an uncertainty value by measuring the distance between the farthest landing point the center point. Using kinematics, we were able to calculate the initial velocity of the marble.

Calculations

Because the marble was launched horizontally, we know that the angle in which it was launched was at 0 degrees.

This means that the velocities in the x and y direction are:

To calculate for the initial velocity, we first had to calculate time. We did this by using a kinematics equation in respect to the y component of the trajectory.



Since our marble was launched horizontally, we know that the velocity in the y direction is 0. The acceleration of the marble is due to gravity. The height of the ball is delta y.



Once we had our value for t, we used a kinematics equation in respect to the x component to solve for the initial velocity.



We know that the acceleration of the ball in the x component is 0



Measured Data





Initial Velocity: 1.619313504 m/s.

Part 2: Determine where the marble will strike an inclined board

Procedure:

We added a board such that one end touched the end of the table, and the other end on the floor (like the figure above). To ensure the board did not move, we added weights to the end of the board and used duct tape on the weights.We then launched the ball, noted where the ball landed on the board, and placed a new piece of paper with carbon paper on top. Once our apparatus was set, we launched the ball five times and measured the distance d.

Calculations

To mathematically find the distance along the wooden board, we expressed distance in terms of its x and y components





We then used the same kinematics equations as part one, starting with the x component. Using this kinematics equation, we solved for t.



We plugged our value of y into our kinematics equation for y
'




We then solved for d



Measured Data

Conclusion

Through the mathematical approach, we calculated that distance d was 0.938339123 m, and our experimental distance was within that range. There were a few sources of error within our calculations. For part one, our landing points were more scattered than we anticipated. We ran the experiment several times to receive more accurate data. Our final landing points were relatively closer than our initial landing points, and our calculated uncertainty was 1.7% for the final points. We encountered the same issue in part two of our experiment, and had a calculated uncertainty of 2.0%.

Our main reason for our uncertainty was due to the impact point on the two v-channels. Instead of smoothly transitioning from one rail onto the next, we noticed the marble would hit the second v-channel potentially causing it to lose some velocity and land very scattered in relation to the other test cases. However, we found that even with these uncertainties our experimental values were still reasonable and within our mathematical values. 

The intersection of both v channels