Projectile Motion Lab Report PDF

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Summary

The kinematics of projectile motion in the earth's gravitational field will be studied to gain an understanding of horizontal range, maximum height, time of flight, and trajectory of the projectile....

Description

Benjamin Kelley PHYS.1410 (Physics 1) Section 809

Instructor: Chaminda Ranathunga Date of Experiment: 10/16/18 Partners: Jeffrey Hyde

Experiment:

Projectile Motion

Objective:

Gain an understanding of the aspects of projectile motion such as horizontal range, maximum height, etc.

Introduction

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This lab studies the case of 2D projectile motion. In such a case, horizontal and vertical motion act independently since there is no acceleration in the x direction and there is a constant acceleration of g (9.8 m/s) in the y direction. The motion of an object in such case can be determined by the launch angle of the projectile and the initial velocity. The equations that describe the motion in the x and y direction are as follows: X direction

Y direction

ax=0

ay = -g

vx= constant = vocos�

vy= vosin� - gt

x = (vocos�)t

y = (vosin�)t - gt2/2

We can find equations that calculate various aspects of the projectile by combining these x and y position equations together in various ways. By combining the x-position and y-position equations, we obtain an equation for the trajectory of the object: y = xtan� - (gx2)/2(vocos�)2 By setting y=0 in the y-position equation, we see an equation for the total time travelled by the projectile: t = (2vosin�)/g We can find the total distance in the x-direction (R) by substituting the time equation into the xposition equation: R = (vo2sin2�)/g And finally we can find the maximum y-position of the projectile by substituting half of the time equation into the y-position equation: ymax = (vosin�)2/2g

Apparatus and Procedure Equipment used:

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● Projectile (metal ball) ● Spring loaded projectile launcher ● 2 photogates ● Time-of-flight sensor pad ● Lab jack ● Horizontal scale (meter sticks) ● Vertical scale (2-meter stick) ● Projectile grapher ● Motion sensor ● Tracer paper

Measurement Apparatus

Graphing System Apparatus

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Description of Experiment: The first part of this experiment is to record values for vo at varying powers (“clicks”) of the projectile launcher by setting the projectile launcher at an angle of 0o and launching the projectile. After this is done, set the launcher was set at various angles (10, 20, 30, 45, 60, 70, and 80 degrees) and start the computer program. Launch the ball, and if the ball lands on the scale, the time of flight will be shown. Record this time of flight (t). Measure from the launcher to where the ball left a mark on the carbon paper (this is where the ball landed). Record this distance (R). Finally, set up the 2-meter stick halfway between the calculated R and the projectile launcher and launch the ball again at the same angle. Measure the height at which the ball passes the 2-meter stick. Record this distance (ymax). All of these values are recorded into a table labelled “Table 1”. For the second part of this experiment, set the launch angle of the projectile launcher connected to the large graph to 70o. Launch this ball and record its position on the graph at x=10 cm. Repeat this process and now mark the position of the ball at x=5 cm. Continue the process in this fashion (x=20 cm, then 15 cm, then 30 cm, then 25 cm, etc.) until you have a full curve. Finally, record these (x,y) coordinates into a new table (Table 2).

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Results and Analysis Page 5: Table 1 Page 6: Table 2 Page 7: Graph of Trajectory at 70o Page 8: Analysis

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Analysis After completing the measurements of flight time (t), total change in the x-direction (R), and maximum height (ymax) of the ball, calculations were made to predict said values using the equations from the introduction. Every ball launched in the trials was fired with a power of three “clicks” from the launcher, meaning the ball's initial speed was 5.21 m/s. The equation used for flight time was t = (2vosin�)/g, with almost all calculations falling within 3.78% of the measured values. The only outlier was the calculation for when � = 10o, which had a percent error of 15.87%. The equation used for the total change in the x-direction was R = (vo2sin2�)/g, with almost all calculations falling within 4.22% of the measured values. Once again, the only outlier was the calculation for � = 10o, this time yielding a percent error of 13.68%. Finally, the equation used for the maximum height was ymax = (vosin�)2/2g, with almost all calculations falling within 2.87% of the measured values. For a third time, the only outlier was the calculation for � = 10o. However, this time only yielding a percent error of 5.00%. For the graphing portion of this lab, y-coordinates were measured by visually seeing the ball’s position at a certain x-coordinate when launched at an angle of 70o and an initial velocity of two “clicks”, or 3.083 m/s. To calculate the y-coordinates, the equation y = xtan� - (gx2)/2(vocos�)2 was used. The calculated y-coordinates had ymax at a different xcoordinate than the measured one, a different ymax value than the measured value, and a negative y-coordinate at x = 0.65m. Possible reasons for these differences will be brought up in the discussion portion of the report. An analysis requested by the lab manual calls for the creation of this the graph that follows:

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Analysis (continued) This analysis requested a plot with three separate curves all on the same graph. The graph contains a curve representing Calculated Range vs Launch Angle (R vs ), Calculated Maximum Height vs Launch Angle (ymax vs ), and Calculated Flight Time vs Launch Angle (t vs ). This graph allows us to see the relationship each value (R, ymax, and t) has with . By observing the R vs curve, we can see that these two variables have a quadratic relationship, with R climbing to some maximum as increases and falling back down to 0 as continues to decrease. By observing the ymax vs curve, we can see that these two variables have a cubic relationship, with ymax always increasing over , but having some point of inflection somewhere on the curve. Finally, by observing the t vs curve, we can see that these two variables have a radical relationship resembling that of y = x0.5 where t is increasing over , but at a decreasing rate.

Discussion After completing the experiment and analyzing the data, it appears that the equations that exist for finding the trajectory of an object with 2D projectile motion can accurately predict the different aspects of said object, such as flight time, range, etc. One of the only uncertainties that remains from the experiment is the cause of the large percent error for values found at 10o . These large percent errors could be related to the fact that calculations were fairly difficult to make when the ball was launched at an angle of 10o. Measuring range and max height were difficult due to the large portion of horizontal velocity the ball had. The other uncertainty that remains is the cause behind the many differences between the measured y-coordinates and the calculated ycoordinates for the graphing portion of the experiment. One thing that may have caused these differences is a possible portion of velocity happening in a third dimension. With this possible

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third portion of the velocity not being accounted for, this could lead to major differences in the measured values vs the ideal calculated values. Other than these two uncertainties, this lab ran very smoothly.

Conclusion In conclusion, we can confirm that the equation y = xtan� - (gx2)/2(vocos�)2 properly predicts the trajectory of an object with 2-dimensional projectile motion. This means that only the initial velocity and launch angle of an object are needed to predict its trajectory. In addition to confirming this, we now know that there are mathematical relationships between various aspects of the objects trajectory and the launch angle of the object. These mathematical relationships include a quadratic relationship between range and launch angle, a cubic relationship between maximum height and launch angle, and a radical relationship between flight time and launch angle.

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Questions 1. For an initial velocity of 15 m/s and a launch angle of 55 degrees, where will the projectile strike the side of the hill (which makes a slope angle of 20 degrees)?

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This means that the projectile will hit the hill when it has reached at a distance of 16.076 meters away from where it was launched. We can then find the height of the object with this x value:

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So, this means that the projectile will strike the side of the hill at a height of 5.851 meters when it is 16.076 meters away from where it was launched....