14-Sept-2016: Trajectories

PURPOSE:

We must use our knowledge of projectile motion in order to predict the landing point of a ball upon an incline.

INTRODUCTION:

Picture a ball rolling along a table and falling off an edge onto the floor. This is essentially the situation that our group was presented with. With this in mind, we are trying to measure the horizontal and vertical distance traveled by a ball in projectile motion. Given these measurements, we shall use kinematics equations to calculate the initial horizontal velocity of the ball. The initial velocity will give us insight into the ball's trajectory. The question now becomes:

If we were to place an incline of angle theta in the trajectory of the ball, where would the ball land along the incline? 

In other words, picture the same ball rolling along a table and falling off the edge onto a ramp. We must predict the ball's point of impact along this ramp. To do this, we must derive an equation that illustrates a relationship between the initial horizontal velocity of the ball and the distance, d traveled along the incline.

APPARATUS: PART 1:

1) We set up the apparatus as shown:

Diagram of apparatus

1) Secure Aluminum "v-channel" #1 to the table. We kept ours in place with lumber, metal weights, rails, and some tape.

2) Line up Aluminum "v-channel" #2 such that it flows into the the first "v-channel" at an angle. Make sure that the angle of incline for the second "v-channel" is not too steep to ensure the steel ball does not bounce when it hits the transition between the two channels.

3) Line up the ring stand with Aluminum "v-channel" #2 and use the clamp to secure them together. In reality, we just secured the clamp to the ring stand and allowed the "v-channel" to rest on top of the clamp. Adjust the angle of incline as needed.

4) When you are done, the apparatus should look something like this:

Fully assembled apparatus

PROCEDURE: PART 1:

1) Launch the ball from a point near the top of the inclined ramp. We marked our starting point with tape to keep it consistent. Notice where it hits the floor.

2) Tape a piece of carbon paper to the floor where the ball landed. We placed a blank piece of paper underneath the carbon paper so it would have a surface to mark on afterward. 

Carbon paper secured to floor

3) Launch the ball five times from the same place you launched it initially. Ensure that the ball lands in virtually the same place every time. 

4) Determine the horizontal and vertical distance the ball traveled during its launch. We measured these distances with a meterstick. We also included our uncertainty in each value:

5) Determine the launch speed of the ball using the measurements you obtained. In this case, we utilized the average delta x value for our calculations. Note: Don't forget to convert from centimeters to meters first. Here are our calculations: 

Launch speed calculations

APPARATUS & PROCEDURE: PART 2:

6) Imagine attaching an inclined wooden board to the edge of the table such that now the ball, launched from the same place as before, will strike a distance d along the board, like so:

7) Derive an expression that would allow you to solve for d given that you know Vo and alpha. Note: I used theta instead of alpha to signify the angle measure. Additionally, don't round any of your intermediate values--e.g. Vo and theta. Here is our derivation of this expression with its corresponding calculation of d:

Note to self: Use pen next time! Too light!

7) Now it is time to test our predictions. Place a wooden board such that it touches both the edge of the table and the floor. This will act as a ramp for the ball as depicted in the previous diagram. Once you have reasonably adjusted the slope of your ramp, secure it to the floor by pushing some weights up  against it and taping them to them ground. Finally, measure the angle of the board. We used an iPhone app to make this measurement. This is how ours looked:

8) Attach a piece of carbon paper to the board such that it is within the vicinity of where you predicted the ball would land. Now run the experiment, launching the ball five times from the same spot. In our case, we only had enough time to run it once.

9) Determine the experimental value of your landing distance d with its corresponding uncertainty. For example, according to my group-mate, our experimental landing distance d was 75.0 ± 0.1 cm. Here are our final findings:

CONCLUSION:

It appears that our theoretical calculations were fairly close to our experimental values. In fact, our predicted value for the landing distance d was less than 4 centimeters away from the one we measured experimentally. However, this does not hide the fact that there are a number of factors that have contributed to the inaccuracy of our measurement of d. One of the most prominent causes that come to mind is human error. This is because there will often times be a misreading of measurements from tools such as rulers and calipers. The marking are small, and minute changes in perspective can cause some group members to see one measurement and others group members to see another; such was the case with our group from time to time. Additionally, there were certain assumptions we had to make about the world that could have affected the accuracy of our calculations. For example, when carrying out our calculations for d and Vo, we assumed that acceleration due to gravity equals -9.8 m/s². In order to be more precise, we could have utilized the value -9.81 m/s² instead and perhaps increased the accuracy of our calculations. Lastly, if our group had run more trials at the end of the experiments, we could have gotten a more accurate understanding of the experimental value of the landing distance d. With an average of these trials, we also could have a better grasp on how accurate our theoretical values were in comparison to the values we measured.

GROUP MEMBERS: Matthew I., Xavier L., Billy J.