Projectile Motion Lab
Partners: Madeline Walbert and Maeann Brougher
Date: 10/28/14
Purpose:
In this lab, we will determine the initial velocity and the range of a spherical projectile shooting out of a cannon.
Date: 10/28/14
Purpose:
In this lab, we will determine the initial velocity and the range of a spherical projectile shooting out of a cannon.
Theory:
In order to find the range, the equation shown in Figure 2 had to be derived. The quanties associated with the derivation are:
Range (R)=Total distance traveled by the projectile.
Initial Velocity of X (Vox or VocosƟ)=Starting rate at which the projectile changes its position horizontally, or in the X direction.
Initial Velocity of Y (Voy or VosinƟ)=Starting rate at which the projectile changes its position vertically, or in the Y direction.
Initial Velocity (Vo)=The Velocity at which the projectile is launched.
Time (t)=Period in which the projectile moves.
Height (Δy)=Initial distance from the ground to the projectile.
Gravitational Acceleration(g)=Force acting on projectile.
Theta (Ɵ)= Angle in which the projectile is launched.
The initial velocity and height had to be measured and calculated first before solving the predicted range.
Range (R)=Total distance traveled by the projectile.
Initial Velocity of X (Vox or VocosƟ)=Starting rate at which the projectile changes its position horizontally, or in the X direction.
Initial Velocity of Y (Voy or VosinƟ)=Starting rate at which the projectile changes its position vertically, or in the Y direction.
Initial Velocity (Vo)=The Velocity at which the projectile is launched.
Time (t)=Period in which the projectile moves.
Height (Δy)=Initial distance from the ground to the projectile.
Gravitational Acceleration(g)=Force acting on projectile.
Theta (Ɵ)= Angle in which the projectile is launched.
The initial velocity and height had to be measured and calculated first before solving the predicted range.
Experimental Technique:
Initial Velocity
Figure 3: Photogate
The initial velocity was measured by setting up a photogate timer onto the cannon. Once the ball shot out of the cannon, it passed through the two gates. The velocity in between the gates was then measured through DataStudio on the computer. In order to calculate a more accurate initial velocity, the projectile was shot 10 times and the average of the 10 velocities was by adding all the shots together and dividing it by 10.
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Range
Figure 4: Carbon Paper
The equation was derived simply by taking what values equaled and substitution into what other values equaled. Time was found to equal a quadratic equation, which was then plugged into was range equals so that one long derived equation can be solved to find range. In order to use the derivation, height had to be measured. A plumbob was held against the center of the projectile on the cannon and a mark was made on the floor where it hit. Measuring tape was put up against the length of the plumbob line that went from the projectile to the floor and was measured. The angle at which the projectile is launched at is already known because the cannon was set to a specific angle. After plugging in all the values into the derived equation, a predicted range was solved. This predicted range was then measured out on the floor from the dot placed earlier on the index card with the plumbob. A piece of paper was placed in the area of the predicted range and a line was marked going across it at where the range was calculated to be. Carbon paper was placed on top of the white paper and 10 shtos were fired, hitting the carbon paper which then went through and made black marks on the white paper. Each black mark, or shot, was then measured from the index card using measuring tape. Each measured shot was then added together and divided by 10 to receive the average range of the projectile. Laslty, the predicted range had to be compared to the actual range by calculation percent difference. Percent difference is the absolute value of the predicted value minus the actual value divided by the actual value, and then multiplied by 100.
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Data and Analysis:
Initial Velocity
Figure 5: Average Initial Velocity
After the 10 shots were fired, the average initial velocity was calculated out to be 3.12 m/s.
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Figure 6: Analytical Projecitle Diagram
Figure 7: Analytical Derived Equation
After each quantity was known and plugged into the derived equation, the range solved out to be 1.63m in front of the cannon and -0.702m behind the cannon. The actual range values of the 10 shots fired were 163.25cm, 163.50cm, 164.35cm, 164.75cm, 164.80cm, 164.85cm, 165.25cm, 165.60cm, 165.30cm, and 165.60cm. Those values were added together and divided by 10 to receive an average range of 164.725cm, or 1.65m. The percent difference was then solved by taking the absolute value of 1.63m-1.65m/1.65m and then the whole thing was multiplied by 100 to receive a percent answer. In the end, the percent difference was 1.21%.
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Conclusion:
In the lab, a projectile's initial velocity was found and a range was predicted using a derived equation. The actual range was found by shooting ten shots onto carbon paper and measuring each one to solve for the average range. The actual range was then compared to the predicted range by using the percent difference equation, which came out to be a 1.21% difference. In other words, the predicted value found with the derivated equation was 1.21% different than the actual value of the range. The uncertainty in the range measurement would be 1.63 ± .01 m for the predicted value and 1.65 ± .01 m for the actual value.
Obviously since there is not a 0% difference between the predicted and actual ranges, there is error involved. One error I know my group conducted was that we did not measure the height to the bottom edge of the projectile, we measured to the center. It makes sense to measure to the bottom edge because that is what will be hitting the ground up on impact. Even though that will only change the height answer a little bit, it will sitll end up change the predicted value for range in some way. Another way I could have recieved a better range prediction is if I kept more decimal places when I got an answer. This could have then been used to compare more precisely to the actual range, since I got that with more decimals. The last error is another measurement error. When measuring each individual actual range, I could have easily assumed the last decimal place wrong, since that place is not easily determined. There is definitely uncertainty with the hundreths place. Lastly, I believe wind resistance is a factor. The projectile was shot through the air, meaning it hit and smacked particles and molecules while flying through the air. This collision surely caused some friction and setback, but it was not that major since the ball was not going at that fast of a speed.
Obviously since there is not a 0% difference between the predicted and actual ranges, there is error involved. One error I know my group conducted was that we did not measure the height to the bottom edge of the projectile, we measured to the center. It makes sense to measure to the bottom edge because that is what will be hitting the ground up on impact. Even though that will only change the height answer a little bit, it will sitll end up change the predicted value for range in some way. Another way I could have recieved a better range prediction is if I kept more decimal places when I got an answer. This could have then been used to compare more precisely to the actual range, since I got that with more decimals. The last error is another measurement error. When measuring each individual actual range, I could have easily assumed the last decimal place wrong, since that place is not easily determined. There is definitely uncertainty with the hundreths place. Lastly, I believe wind resistance is a factor. The projectile was shot through the air, meaning it hit and smacked particles and molecules while flying through the air. This collision surely caused some friction and setback, but it was not that major since the ball was not going at that fast of a speed.
References:
Bowman, D. (n.d.). Lahs Physics. Lahs Physics. Retrieved October 29, 2014, from http://lahsphysics.weebly.com/