Lab 1.5 Report PDF

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Lab report for physics 1301W ...

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Lab I, Problem 5: Motion Down an Incline with an Initial Velocity

Abstract The correlation between initial velocity and acceleration were studied using a cart on an angled ramp and video analysis software. A cart was pushed with an initial velocity down the ramp and its position and velocity versus time graphs were computed. It was found that the initial push did not affect the acceleration versus time graph, thus, initial velocity has no effect on acceleration. Introduction: A company that is designing a new bobsled for the U.S. Olympic team wants to know how the initial push of the sled affects the motion throughout the course.1 They know that the speed depends heavily on the quality of the initial push, but they want to know the correlation between initial velocity and acceleration in particular.1 In order to determine this correlation, the situation will be modeled by pushing a cart down an inclined track and analyzed using video software. Predictions: The effect of initial velocity on an object’s acceleration can be calculated by using calculus and simple reasoning. The acceleration with respect to time of an object is calculated by taking the derivative of its velocity versus time graph. For example, if the velocity of the cart can be modeled by the equation y=x +5, the acceleration graph would simply reflect the equation y=1 because the derivative with respect to x of x equals 1, and the derivative of a constant (5) is 0. In a simple equation such as this, the constant represents the initial push and equals the velocity at time=0. However, the derivative of an equation only reflects the slope, and therefore the constants are irrelevant. Using this knowledge and simple reasoning it is possible to conclude that the initial push of the bobsled will have no effect on the acceleration, and the acceleration will look the same in our trials because the only factor affecting it is gravity. Shown below are the general form of the equation for a quadratic position versus time graph and its first and second derivatives which prove that the acceleration value is not dependent on the velocity. Further below is the correct prediction for this equation, and the prediction used during the lab which contained an error that will be discussed later. 1 2 p(t)= at + vt + p , where a= acceleration, v= initial velocity, and p= initial position 2 at + v , p″ (t )= v ′(t )=a (t )= a p′ (t)=v (t )=¿

2 2 83.25 ¿ 83.25 ¿ +0.1 v +0 p 1 +0v+0p, with no initial velocity or 1 Correct prediction: with p(t)= ¿ p(t)= ¿ 2 2 some initial velocity 2 2 83.25 ¿ +0 v +0 p 83.25 ¿ +0.1 v +0 p with no initial velocity, or with Predictions used: p (t)=¿ p(t)=¿ some initial velocity

Procedure: A ramp was propped up with a wooden block at one end to make an inclined track for the cart. The height of the incline was 17.3 ± 0.05 cm and the length of the ramp or the hypotenuse of the triangle made was 130.0 ± 0.5 cm. The acceleration due to gravity that would affect the cart was calculated by taking into account the fact that the axes were rotated from what would be considered standard, as seen in the image below. A meter stick was held against the hypotenuse while the video was taken and used to calibrate the video analysis software. The software was calibrated to view the angled ramp as the x-axis.

y x

17.3 cm 7.65°

The frame of the video was set up to record the latter portion of the cart’s path because after trial runs, it was found that omitting the beginning of the run resulted in a more consistent graph. A video was then taken of the cart traveling down the track without being pushed, and then a second was taken with an initial push (thus, with an initial velocity). After importing the videos to MotionLab, position versus time and velocity versus time graphs were made. Data: Total distance traveled by cart: 130.0 ± 0.05 cm Height of ramp:17.3 ± 0.05 cm Angle of ramp: 7.65 ± 0.025° (calculations below) 17.3 ± 0.005 𝜃=sin 130.0 ± 0.005 ) ′¿

R=( X ⋅Y )/Z δ

δ R=|R |√ ❑

17.3 ±0.005 = (0.133) √ ❑ =0.133*0.003=3.99*10-4 130.0 ±0.005 𝜃+=sin′(0.133+3.99*10-4)=7.67° 𝜃-=sin′(0.133-3.99*10-4)=7.62° 𝜃=7.65±0.025°

(Ideal) Acceleration due to gravity with rotated axes: 83.25±0.25cm/s² (calculations below) Acceleration when changing the y component (such as using a ramp): θ ) g ¿ 9.8( 90 ° g+ =83.5 cm/s2 (calculated using +) g-=83.0 cm/s2 (calculated using -) g=83.25±0.25 Using the data listed above, the predicted equation for the run without an initial push was p(t)=83.25 t 2 +0 t . The predicted equation for the run with an initial velocity was p(t)=83.25 t 2 +100 t . The manually fitted equations of the aforementioned trials after data was collected were

p(t )=73.3 t 2 +0 t

and

p(t)=73.3t 2 +85 t

respectively.

From the position vs. time graphs, the computed derivative of the run without an initial push is dp dp =146.6 t , and the derivative of the run with an initial velocity is =146.6 t+100 . dt dt After deriving these equations once again to find the accelerations, they were both found to be 146.6

cm . 2 s

Analysis: Manually Fit Position vs. Time - No Initial Push

time (s)

p(t )=73.3 t 2 +0 t

Manually Fit Position vs. Time - With Initial Push

time (s)

p(t)=73.3t 2 +85 t

As seen by the equations of graphs above and their derivatives mentioned earlier, the velocity of a cart on a ramp is affected by an initial push, however this does not affect the acceleration. When the position versus time graph is derived two times, the only coefficient remaining is the magnitude of acceleration. The calculation for the ideal acceleration due to gravity treating the hypotenuse as the x-axis is 0.8325m/s² or 83.25cm/s². However, when deriving the equations above, the magnitude of the acceleration due to gravity would be almost two times the ideal value. When deriving the prediction equation twice and finding the acceleration, it is equal to twice the magnitude of the ideal acceleration. p(t )=73.3 t 2 +0 t , p′ (t)=146.6 t , p ″ (t)=a=146.6 cm/s2 2 83.25 ¿ +0 v +0 p Prediction equation and derivatives: , p″ (t )=a=166.5 cm/s2 p (t)=¿ Ideal acceleration: 83.25 cm/s2 (56% of experimental value and 50% of predicted value) This is due to a mistake made when accounting for the fact that in the analysis software, the 1 a or half of the acceleration, so when the coefficient in front of the t² value corresponds to 2

derivative is taken, the coefficient should double. Because of this, the prediction equations should have had a coefficient of 41.65, giving an acceleration of the ideal 83.3cm/s². Although the accelerations differed by a factor of two which is very large, the graphs still appeared to fit the prediction. This outcome was suspected to be because the length of the track was not very long, which resulted in a small graph where the rate of increase of the slope was not very prominent. Despite this error, the most substantial finding from these trials is the fact that the acceleration remained constant regardless of an initial velocity. The amount of error in calculating the acceleration and comparing it to the ideal value was very large (76.1%) because of the previously mentioned mistake when accounting for the coefficients in the kinematics equations. Any other errors that could have accounted for the discrepancy between the ideal acceleration and the acceleration found from the graph are human error by measurements, the use of MotionLab, or the fact that the table upon which the track was set up was not completely level. The uncertainty in the measurements of length and height of the track are aforementioned and this contributed to uncertainty in the value calculated for theta. These calculations can be found in the data section of the report. Furthermore, it is probable that there was a small amount of error in calibrating the video analysis software. The meter stick against the ramp could have been slightly lower than the track itself, making the calibration slightly off, or there could have been discontinuities when choosing a point on the cart during the video analysis process. In addition, the kinematics equations used presume that there are no external forces such as air resistance or friction, which also would have impacted the final acceleration. As previously mentioned, it is impossible to calculate the percent error due to the calculation mistakes, however had the acceleration been measured correctly, all these sources of error would have been accounted for. Conclusion: It was concluded that the initial velocity of an object does not have an affect on its acceleration. The position versus time graphs yielded a consistent value for acceleration regardless of initial velocity. Although not accurate regarding the actual acceleration due to gravity because of the calculation error stated previously, the value of acceleration was not dependent on the speed of the cart before recording the videos. These results proved the prediction correct. These findings signify that the U.S. Olympic bobsled team1 only needs to account for the initial velocity when determining the position with respect to time or the velocity with respect to time. They do not, however, need to account for the initial push when calculating acceleration down the track.

1) School of Physics and Astronomy. Lab handbook. University of Minnesota. Minneapolis, Minnesota. 2018. Digital. As seen in the graph above, the displacement of the electron beam increases linearly when the strength of the electric field in the CRT is increased. Based on the equation in the prediction section, this is fairly obvious but well justified. When an electron passes through an electric field it moves in the direction of the field, or toward the positively charged plate. If the difference between plates is greater, or the negative plate is more negative and vice versa for the positive, intuitively the electron will feel more repulsion from the negative side and be more strongly attracted to the positive one. It can be seen that the majority of the experimentally recorded data points were close in proximity to the theoretical calculations due to the fact that the theoretical tend to be within the error bars....