Materials:
- Aluminum "v-channel"
- Small Steel Ball
- Wooden Board
- Ring Stand
- Clamp
- Carbon Paper
- Tape
Setup:
To set up the experiment, we had to make an apparatus that would allow the steel ball to roll off our table with some initial velocity. To do this, we used and connected to aluminum "v-channels." One of the v-channels would be horizontal and parallel with the surface of the lab bench, while the other would be inclined and connected roughly on the halfway point of the horizontal "v-channel." To get the incline, we secured the second "v-channel" to the ring stand using a small metal pole and tape. The edge of the horizontal "v-channel" would be in line with the edge of the lab bench. The final setup can be seen in Fig. 1-1 and Fig. 1-2.
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| Fig. 1-1 We ensured the velocity of the ball would be the same by marking the point at which we would release the ball. |
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| Fig. 1-2 The ring stand was used to support the inclined "v-channel." |
Making Our Prediction:
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| Fig. 2 We had to drop the ball several times to get a general idea of where it would land. |
To make a semi-decent prediction of where the ball would land on an inclined board, we first needed to determine the initial velocity of the ball is it fell off the "v-channel." To do this, we would need to measure how far the ball would land horizontally from the edge of the "v-channel." To do this, we first let the ball go from a marked position on the inclined "v-channel," and allowed it to fall of the edge of the lab bench. Once we had a rough idea of where the ball would land, we placed a piece of carbon paper onto the floor in that general area (see Fig. 2). The carbon paper would have a mark indicating how far the ball fell. We let the ball fall several times. We then measured the distance from the parked point to the edge of the table. This was our horizontal ("x") displacement. (The ball landed in a certain range of locations, we used the average distance for our calculations). We then measured the height of the fall, which was the vertical distance from the table to the floor. At this point, we used kinematics to find the initial velocity of the ball. (see Fig. 3-1 and Fig. 3-2). The initial velocity of the ball was 1.079 m/s in the horizontal direction.
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| Fig. 3-1 Data Table |
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| Fig. 3-2 We could find time be using kinematic equations relative to the y-axis. We could then plug-in time into our x equation to find the initial velocity. |
Once we had our initial velocity, it was time to make our prediction. We had to imagine that there was now an inclined board that connected to the edge of the "v-channel." This board would be at some angle "α". The distance at which the ball would hit the board at would be considered "d". Our goal now was to derive an expression that would allow us to calculate "d", given initial velocity and the angle "α". Once we had this expression, we measured our last unknown "α", which was 49.1 degrees. Our derivation and prediction can be seen in Fig. 4. Our prediction was that the ball would land 0.4190 m down the inclined board.
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| Fig. 4 We used our kinematic equations to derive an expression for our predicted distance "d". |
We finally set up the board and ran the actual experiment. The set up of the inclined board can be seen in Fig. 5. Again, we used carbon paper to find a fairly accurate spot at which the ball landed on the board. After letting the ball fall several times, we found that on average, the ball landed 0.4285 m down the ramp. We found our calculated value to be -2.22% off from the actual value. This indicated that we had made a good prediction.
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| Fig. 5 We finally set up our inclined board and once again placed carbon paper in the general area the ball was landing. |
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| Fig. 6 Once we had our value, we had to calculate the combined uncertainty in both our calculated value of "d", and our actual measured value of "d". |
Our uncertainty:
We used partial derivation to find the uncertainty in our predicted value of "d". (see Fig. 6). Our uncertainty was roughly 0.88 cm. Thus our final predicted value of "d" was 41.90 cm +/- 0.88 cm. The "actual" value of "d" was measured to be 42.85 cm. At first glance, it may seem as if we are not within that range, even with our certainty which yields us a maximum value of 42.78 cm. However, we must consider that our "actual" value of "d" had a range that left it with an uncertainty of 0.45 cm. Thus, our data may be considered acceptable, so long as we show our possible error.
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