Lab 5: March 15, 2017: Trajectories
Lab 5: Trajectories
Amy, Chris, and John
March 15, 2017
Today's lab was to use our understanding of projectile motion to predict the impact point of a ball on an inclined board, and compare our experimental and calculated values.
Theory:
Theory:
This lab seeks to predict the impact point of a marble launched horizontally off of an aluminum "v-channel" . By testing to see if the marble lands in virtually the same spot through a number of tries (five in our initial test) we can measure the height and distance away from the launch point. This allows us to apply our constant acceleration kinematic formulas to calculate for the initial velocity (launch speed) of our ball.
With this information, we can calculate for the value which the experiment wants to compare, the impact point of the marble on a wooden board on an incline. We solve for the distance from the launch point to the impact point mathematically, and we run the experiment as well so we can compare the two values.
All of the procedure mentioned above solved mathematically looks like this:
We know that at launch from the ramp, the marble is horizontal so
from the horizontal
this means the x and y components from initial velocity are:
We first solve for t using:
where y-component of initial velocity is 0, and a = gravity = 9.8 m/s^2, and delta y is the height.
We plug this t back into the delta x equation to solve for x-component of initial velocity which as shown above is just the initial velocity:
in this equation, a = 0 since there is no acceleration in the x direction, only in the y (by the force of gravity). So this above equation simplifies into:
and this calculates from initial velocity.
For the distance along the wooden board, we express the x and y components of the distance with the angle alpha. Then we derive an expression for d (distance) in terms of initial velocity and alpha (the angle above the floor).
From kinematic equation for y:
we know y component of d is d sin alpha so:
Experimental Procedure:
The data we obtained from part 1 was:
Also from the formula in theory/introduction, the calculated distance value was 0.938339123 m.
This expression allows us to solve the value for d mathematically.
Part 1:
We set up our experiment on the table using a ring stand with a clamp, and 2 aluminum v-channels, one angled downwards from starting point which slides into the second ramp that goes horizontally. We placed a couple piece of wooden blocks with indents that holds the horizontal v-channel in place.
For each run, the marble starts from rest at the same place on the inclined v-channel to make sure we get as close to the same conditions every run. A piece of paper is taped to the floor where we expect the marble to land and its covered by carbon paper. The carbon paper makes markings on the taped paper that will record the impact point. After doing this 5 times, we can measure with a meter stick the horizontal distance and assign an uncertainty value (we eyeballed the "center" point of our scatter of dots and took the furthest distance away from it as our uncertainty). The height is calculated using a string with a weight attached to allow us to make an accurate measurement by measuring the straightened string.
Mathematical Model:
With this info, we calculate for launch speed (shown above in the theory/introduction) and run the second part of our experiment, which is very similar to the first but with a wooden board on an incline from the launch point. The angle from the floor was measured using an iPhone app and we first develop a mathematical model for this distance (also shown above in the theory/introduction).
Part 2:
Afterwards, we run the experiment with the board very much like we did in our first portion (with an altered setup shown in Fig. 4). We then measure with the meter stick the distance along the board from the launch point and assign an uncertainty value just like we did for the first part of our experiment (the paper on the board shown in Fig. 5).
Finally, we compare the mathematically obtained value with our experimental value and analyze any potential sources of error and causes of any potential discrepancies in the two values.
Measured Data:
Our calculated distance along the wooden board for part 2:
Calculated Data:
From the formula in the theory/introduction, we obtained an initial velocity of 1.619313504 m/s.
Conclusion:
The experimental distance value we obtained from our mathematical solving was within that that range. There were quite a few sources of error that introduced uncertainty in our measurement, mainly the distance away from table. For our x component of impact point, we found our dots were more scattered than we had expected yielding bigger uncertainty values. We average the distance out and used that in our final calculations. For the first part of the experiment, our uncertainty was 1.7% which is within our accepted range of error (2%) but still a relatively large source. Distance along the board also had a relatively large ratio of error yielding 1.95%.The biggest error in impact point we determined was our ramp setup. We were getting slightly different launch speeds off the ramp when the marble rolled from the first v-channel to the second. Instead of a steady roll down the ramp, it would bang along the sides of the ramp into the second rail and lost some speed, that and being launched at slightly different speeds could have resulted in error. We concluded this was the main source of our triangular impact point and our biggest source of uncertainty.
With this information, we can calculate for the value which the experiment wants to compare, the impact point of the marble on a wooden board on an incline. We solve for the distance from the launch point to the impact point mathematically, and we run the experiment as well so we can compare the two values.
All of the procedure mentioned above solved mathematically looks like this:
We know that at launch from the ramp, the marble is horizontal so
this means the x and y components from initial velocity are:
We first solve for t using:
where y-component of initial velocity is 0, and a = gravity = 9.8 m/s^2, and delta y is the height.
We plug this t back into the delta x equation to solve for x-component of initial velocity which as shown above is just the initial velocity:
in this equation, a = 0 since there is no acceleration in the x direction, only in the y (by the force of gravity). So this above equation simplifies into:
and this calculates from initial velocity.
For the distance along the wooden board, we express the x and y components of the distance with the angle alpha. Then we derive an expression for d (distance) in terms of initial velocity and alpha (the angle above the floor).
From kinematic equation for y:
we know y component of d is d sin alpha so:
Experimental Procedure:
 |
| From the lab handout, the first portion of the lab where we find height and distance to calculate for launch speed. |
 |
| The paper taped to floor with carbon paper above it. |
 |
| The second part of lab, where we find the distance along the wooden board. |
 |
| The paper markings for wooden board with weight taped at the end to prevent it from moving. |
Also from the formula in theory/introduction, the calculated distance value was 0.938339123 m.
 |
| The point where the marbled switched rails from one v-channel to the next. |
This expression allows us to solve the value for d mathematically.
Part 1:
We set up our experiment on the table using a ring stand with a clamp, and 2 aluminum v-channels, one angled downwards from starting point which slides into the second ramp that goes horizontally. We placed a couple piece of wooden blocks with indents that holds the horizontal v-channel in place.
For each run, the marble starts from rest at the same place on the inclined v-channel to make sure we get as close to the same conditions every run. A piece of paper is taped to the floor where we expect the marble to land and its covered by carbon paper. The carbon paper makes markings on the taped paper that will record the impact point. After doing this 5 times, we can measure with a meter stick the horizontal distance and assign an uncertainty value (we eyeballed the "center" point of our scatter of dots and took the furthest distance away from it as our uncertainty). The height is calculated using a string with a weight attached to allow us to make an accurate measurement by measuring the straightened string.
Mathematical Model:
With this info, we calculate for launch speed (shown above in the theory/introduction) and run the second part of our experiment, which is very similar to the first but with a wooden board on an incline from the launch point. The angle from the floor was measured using an iPhone app and we first develop a mathematical model for this distance (also shown above in the theory/introduction).
Part 2:
Afterwards, we run the experiment with the board very much like we did in our first portion (with an altered setup shown in Fig. 4). We then measure with the meter stick the distance along the board from the launch point and assign an uncertainty value just like we did for the first part of our experiment (the paper on the board shown in Fig. 5).
Finally, we compare the mathematically obtained value with our experimental value and analyze any potential sources of error and causes of any potential discrepancies in the two values.
Measured Data:
Our calculated distance along the wooden board for part 2:
Calculated Data:
From the formula in the theory/introduction, we obtained an initial velocity of 1.619313504 m/s.
Conclusion:
The experimental distance value we obtained from our mathematical solving was within that that range. There were quite a few sources of error that introduced uncertainty in our measurement, mainly the distance away from table. For our x component of impact point, we found our dots were more scattered than we had expected yielding bigger uncertainty values. We average the distance out and used that in our final calculations. For the first part of the experiment, our uncertainty was 1.7% which is within our accepted range of error (2%) but still a relatively large source. Distance along the board also had a relatively large ratio of error yielding 1.95%.The biggest error in impact point we determined was our ramp setup. We were getting slightly different launch speeds off the ramp when the marble rolled from the first v-channel to the second. Instead of a steady roll down the ramp, it would bang along the sides of the ramp into the second rail and lost some speed, that and being launched at slightly different speeds could have resulted in error. We concluded this was the main source of our triangular impact point and our biggest source of uncertainty.
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