edited Phys1 Lab2

Lab #2: Motion II PHYS-UA11

Objective and Description In this lab, we are able to analyze the relationship between position ( x ), velocity ( v ), acceleration ( a ) and time ( t ) in a physical sense. This is accomplished through the aid of the Capstone database, which this lab allows us to become even more familiar with. As in the previous motion lab, the black motion sensor that uses sound waves to detect the distance of an object was used. By attaching an index card to two different gliders, the motion sensor was able to track the motion of the index card as the glider traveled down an inclined air track as a function of time. Both the incline, 𝜽 , and the size of the glider were manipulated in the experiment. Theory There are several theories that are applied in the functioning and understanding of the experiment. For instance, we could make assumptions about what our graph should look like based on the knowledge of the mathematical relationship of the values. It is known that velocity is the derivative of position and acceleration is the derivative of velocity and therefore the second derivative of position. If acceleration is constant the following equations can be obtained from the integrations: x=x 0 + v 0 t + ½at 2 v=v 0 + at Furthermore, if friction is neglected, we are able to use Newton’s second law to predict that the acceleration of the glider is g •sin 𝜽 . This comes from the fact that 𝚺 F X =ma x . Since there is no friction, the x component of gravity is the only force acting upon the glider. Since mg x =mg•sin 𝜽 :

→ ?𝑖?Θ = 𝑦 ? Θ = ?𝑖? −1 ( 𝑦 ? ) Θ 1 = ?𝑖? −1 ( 𝑦 1 ? ) = ?𝑖? −1 ( 0.0191 1.2857 ) =0.01486 Θ 2 = ?𝑖? −1 ( 𝑦 2 ? ) = ?𝑖? −1 ( 0.0371 1.2857 ) =0.02886 Table 1.2 *Note: sin was selected rather than tan . This is because the track length was used as the hypotenuse and the height of the block was opposite to the angle. Acceleration (m/s 2 ) Combination Trial #1 Trial #2 Trial #3 Trial #4 Trial #5 Average a (m/s 2 ) 𝜽 1 +m 1 a=0.1435 m=0.143 n=1.95 a=0.1429 m=0.127 n=1.85 a=0.1431 m=0.125 n=1.74 a=0.1385 m=0.121 n=1.71 a=0.1375 m=0.123 n=1.79 =0.1411 𝑎 𝜽 1 +m 2 a=0.1432 m=0.142 n=1.81 a=0.1365 m=0.147 n=1.57 a=0.1493 m=0.144 n=1.95 a=0.1499 m=0.148 n=1.72 a=0.1406 m=0.139 n=1.62 =0.1439 𝑎 𝜽 2 +m 1 a=0.2822 m=0.291 n=1.95 a=0.2911 m=0.304 n=1.67 a=0.2936 m=0.293 n=1.76 a=0.2814 m=0.279 n=1.89 a=0.2994 m=0.291 n=2.14 =0.2895 𝑎 𝜽 2 +m 2 a=0.3149 m=0.302 n=1.78 a=0.3744 m=0.320 n=1.80 a=0.2800 m=0.279 n=1.85 a=0.2798 m=0.279 n=1.89 a=0.2843 m=0.285 n=1.93 =0.3067 𝑎 Table 1.3 Acceleration: Theoretical vs. Experimental 𝜽 1 𝜽 2 Theoretical Experimental Theoretical Experimental a=g•sin 𝜽 1 =(9.81m/s 2 )(0.01486) =0.1458 m/s 2 =0.1411 or 0.1439 a=g•sin 𝜽 =(9.81m/s 2 )(0.02886) =0.2831 =0.2895 or .3067

Table 1.4 Part II Figure 1.1 Analysis In this part, the glider was allowed to run down the track, hit the end and bounce back three times. This can best be seen in the velocity graph. From 0 to ~3 seconds, the glider’s velocity increases until ~3.5 seconds, when the velocity goes to zero, representing the first time it hit the end. The glider then begins to move in the negative direction and then forwards again while the velocity increases until ~9.5 seconds where the velocity is zero and the glider hit for a second time. Finally, the earlier pattern is repeated until ~14.5 seconds when the glider hit for the final time. Questions Part I 1. How will acceleration change when we double the mass of the glider? Explain. Based on the fact that F=ma, if mass is doubled, acceleration should theoretically be halved. However, based on the data in table 1.3, when the mass of the glider was doubled and traveled at the same

As demonstrated in table 1.3, after each individual trial, the mean acceleration value, while not identical, was fairly similar to the slope of the velocity plot (m). This further supports what is predicted by the theory. 6. What is your experiential n(power)? What does theory say it should be? Why is there a difference? The n value, as demonstrated in table 1.3, are all approximately 2, either falling slightly below or above, but never quite 2. According the equation x=x 0 +v 0 t+½at 2 , theory suggests that the power should be two, having the shape of a parabola. While all the n values would match this when rounded to the nearest tenth, the slight variance may be attributed to the same reason there is more noise in the velocity and acceleration graphs. 7. When the same 𝜽 is used for different gliders, is there a difference in acceleration? Why? As demonstrated in table 1.3, when the same 𝜽 was used for different gliders, the accelerations, while not identical, were fairly similar. Since a=g•sin 𝜽 was used (see derivation under “Theory”), this observation supports what is expected. Part II 1. Does the velocity curve cross the axis (velocity=0) where you expect it to? The velocity curve crossed the axis each time the glider hit the end of the track, as expected. Additionally, the glider also crossed the axis after hitting the end, when the velocity graph moves from the negative region to the positive region, before dropping to 0 again. This suggests that the velocity was zero as the glider changes direction from moving backwards to forwards. 2. Are the curves the same from bounce to bounce? If not, could you suggest why? The curves become smaller from bounce to bounce. This could possibly be due to the decrease in overall kinetic energy, resulting in less movement and the glider slowing down. Error Analysis

Table 1.4 depicts the difference between the theoretical acceleration found with a=g•sin 𝜽 in comparison to the actual calculated values. It is evident that they two vary only slightly and are generally within an acceptable standard deviation. Ex: % 𝑒???? = (0.2895−0.2831) 0.2831 •100 = 2. 26% Conclusion Through the use of Capstone, the relationship between position, velocity, acceleration and time became more apparent. The application of Newton’s second law in the absence of friction also became an interesting one to observe. Prior to the experiment, I predicted that the acceleration would decrease when we used the larger glider, but this was not observed. Other than that, the experiment matched my expectations and the theory closely.