LAB REPORT 1

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University of Toronto *

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136

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Physics

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Dec 6, 2023

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Abstract In this experiment we determined whether acceleration was constant for an object in freefall, and one in projectile motion, as well as the effects of varying heights and ranges on velocity. For the projectile section, a steel ball was placed in a launcher and projected at 7 varying angles onto a piece of carbon paper. From here the horizonal distance between the landing site of the ball and the launcher was measured, and the theoretical horizontal range was calculated. Air resistance was negligible in this experiment. The free fall portion consisted of a steel ball being dropped from varying measured heights onto a time-of-flight accessory. Average time was calculated to find the acceleration due to gravity. Introduction Projectile motion refers to the motion of an object projected into the air by an initial force, at an angle, where the only force acting on the object is gravity (g). In this situation, air resistance would be negligible, and g would always act downwards (∆x = vxt). The horizontal (x) and vertical (y) components of projectile motion act independently to each other, and are defined by the following: ∆x = usin t ꝋ ∆y = ucos t ꝋ Where u = initial velocity, = angle of projection, and t = time. ꝋ Free fall may be defined as the motion of an object under the sole influence of gravity (g), where the object in free fall would not encounter air resistance. Objects under freefall will always fall at an acceleration equivalent to gravity (a = -g), meaning that regardless of its initial velocity, the acceleration of such an object will always be constant at 9.80m/s 2 . This is proven by the following, to find g: Y = 1/2gt 2 Where g= gravity, and t= time. Procedure Refer to “Free fall and projectile motion” lab manual. Data analysis The freefall section of this experiment was used to determine the acceleration due to gravity of a freely falling object from varying heights. We attached a ball release mechanism to a clamp stand and placed it above a time of flight accessory. The ball was released from an initial height of 0.453m up to a final height of 0.273m at constant intervals of 0.02m, for a total of 10 trials. At each interval we measured the time taken for the ball to hit the time of flight accessory. Referring to graph 1 in appendix 1, the data points define a fairly straight line for all balls except the ones dropped from 0.333m, 0.313m, and 0.293m which give a percentage uncertainty of 0.3%_0.01m within the limits of 6.19%. According to the equation a=-g, our calculated value for g is 4.43ms^2, whereas the accepted value for g is 9.80ms^2. The projectile motion portion was used to determine the acceleration due to gravity of an object projected at an angle to the horizontal. The initial velocities for each angle were calculated by dividing the distance between the photogates by the average time between photogates, these values had a range of 305.31m/s at 20 degrees-274.53m/s at 80 degrees. From here we calculated the horizontal ranges for

each angle using the formula y=ucos ꝋ . A graph for the time of flight versus the y component of initial velocity was also plotted, with a slope of 7.41s. Discussion and conclusions Using the data from graph 1 in appendix one it can be determined that the acceleration due to gravity for a freely falling object is constant. According to the slope of the graph, the value for g is 4.43ms^2, which is approximately half of the generally accepted value of 9.80m/s^2. For the most part, the results presented in appendix one are fairly consistent, however, at the heights 0.333m,0.313m, and 0.293m we can observe a sharp increase in the average time of flight. This does not align with the rest of the results as it deviates from their linear nature and has resulted in a percentage uncertainty of 0.3%. These errors have increased the average time of flight therefore decreasing the value of g. The height also decreases in small increments, I would change this in order to observe changes in acceleration over a larger range of values. According to the “measured horizontal range” table in appendix two, maximum range is produced by and angle of 50 degrees. This may be due to a larger portion of the initial velocity being directed horizontally than say, an angle of 80 degrees (0.122m) where the initial projection is almost vertical. However, whilst similar patterns may be observed, the calculated and measured values for horizontal range do not align for corresponding angles- this may be a result of experimental error when reading measured values, or whilst setting up the experiment. Appendix 1: FREEFALL: TABLE 1: TABLE 1 DATA y(m) t1 t2 t3 t4 t5 t avg t avg ^2 45.3cm 3.16E-01 3.24E-01 3.21E-01 3.25E-01 3.24E-01 3.22E-01 1.04E-01 43.3cm 3.12E-01 3.13E-01 3.14E-01 3.17E-01 3.16E-01 3.14E-01 9.88E-02 41.3cm 3.03E-01 3.07E-01 3.13E-01 3.11E-01 3.10E-01 3.09E-01 9.54E-02 39.3cm 2.99E-01 2.99E-01 3.02E-01 2.99E-01 3.04E-01 3.01E-01 9.03E-02 37.3cm 2.88E-01 2.92E-01 2.96E-01 2.95E-01 2.93E-01 2.93E-01 8.57E-02 35.3cm 2.83E-01 2.86E-01 2.86E-01 2.85E-01 2.84E-01 2.85E-01 8.11E-02 33.3cm 2.77E-01 2.78E-01 2.77E-01 2.79E-01 2.78E-01 2.78E-01 7.71E-02 31.3cm 3.23E-01 2.71E-01 2.74E-01 2.71E-01 2.72E-01 2.82E-01 7.98E-02 29.3cm 2.60E-01 2.59E-01 2.60E-01 2.63E-01 2.63E-01 2.61E-01 6.81E-02 27.3cm 2.50E-01 2.55E-01 2.51E-01 2.53E-01 1.15E+01 2.50E+00 6.27E+00 GRAPH 1:

APPENDIX 2: PROJECTILE MOTION: TIME OF FLIGHT TABLE: Angle Time of flight(s) y component of velocity(m/s) 0 0.136 0 20 0.634 17.9 25 0.214 22.9 30 0.256 28.2 40 0.369 39.4 50 0.448 49.2 65 0.53 64.5 80 0.588 79.9 TIME OF FLIGHT GRAPH: 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time of flight(s)

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